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The exploration of alternative energy sources has gained significant traction in recent years, driven by the urgent need to mitigate greenhouse gas emissions and transition towards sustainable energy. Among these alternatives, CO2 plume geothermal and hydrothermal systems have emerged as promising options due to their potential for providing clean, renewable energy. This study presents a probabilistic investigation into the sensitivity of CO2 plume geothermal and hydrothermal systems under various reservoir conditions in the Williston Basin, North Dakota. In addition to elucidating the impact of reservoir conditions on system performance, the study utilizes probabilistic methods to assess energy output of CO2 plume geothermal and hydrothermal systems. Insights derived from this probabilistic investigation offer valuable guidance for the working fluid selection, systems design and optimization in the Williston Basin and beyond. Results from the sensitivity analysis reveal the profound influence of reservoir conditions on the behavior and efficiency of CO2 plume geothermal and hydrothermal systems. Our case study on Red River Formation and Deadwood Formations shows a potential of 34% increase and 32% decrease in heat extraction based on varying reservoir conditions. Our investigations in the Beaver Lodge field within the Red River Formation yielded arithmetic mean values for CO2 best case resources, hydrothermal resources and the CO2 worst case as 6.36 × 1018 J, 4.75 × 1018 J and 3.24 × 1018 J, respectively. Overall, this research contributes to advancing the knowledge and understanding of CO2 plume geothermal and hydrothermal systems as viable pathways towards sustainable energy production and carbon sequestration. By highlighting the sensitivity of these systems to reservoir conditions, the study provides valuable insights that can inform decision-making processes and future research endeavours aimed at fostering the transition to a low-carbon energy landscape.

Cardiovascular diseases can be caused by numerous beginning conditions. The underpinning causes may include heart/ myocardial inflammation, cardio myopathy, coronary vascular complaint (CAD), dyslipidemia, hypertension, pulmonary hypertension, and arrhythmia, which may eventually affect heart and liver failure. The study was aimed to determine the efficacy, usage and adverse effects of furosemide in diseased heart conditions. Furosemide is prescribed in the treatment of various cardiovascular illnesses, including monitoring the effects in various age groups, gender and body weights. Considering all the primary parameters of drug dose and secondary parameters of dose duration and their certain categories, we evaluate the results based on the effectiveness of furosemide. The collected and gathered data is analyzed by the suitable statistical methods (Central tendency methods) like Arithmetic mean. The effectiveness of furosemide is rapid and strong in the age group of 39-69yrs with the sodium levels of 132.8mEq/L and the urine output of 2.23L/day. The drug shows potential adverse effects like Ototoxicity, skin irritations and stomach cramps, in people with existing heart conditions. The usage of furosemide is beneficial in heart patients however some people show some potential adverse effects. The drug poses its potential side effects only when the usage is high or in unauthorized dose and also depends on previously existed disease conditions.

The aim of this study is to determine the mental toughness levels of male and female football players in sports. The population of the study consists of female football players from the Super League in 2021-2022 and amateur football players in the province of Batman. The sample of the study consists of 232 football players who voluntarily participated in the online questionnaire. Data on the introductory characteristics of the players were obtained with a "personal information form". “Sport Mental Toughness Questionnaire-SMTQ” was used to determine mental toughness of the players. The obtained data were transferred to the SPSS 22 software program for statistical operations. Frequency analysis, percentage analysis, arithmetic mean, t test, Anova analysis and post hoc tests were used to analyze the data. According to the variables of age, education level, years of licenced player and their level of football, statistically significant differences were found between the mental toughness levels among football players (p lt;.05). According to gender, no difference was found between the mental toughness levels of the football players (p gt;.05).

The research aims to identify the relationship of some basic skills with the motor response speed of badminton players of Mosul University. The research community consists of the (30) badminton players from the University of Mosul. The researcher used the following means to collect the data such as a form of recording the speed of response, and the results of the basic skills tests. The data were analyzed statistically using (arithmetic mean, standard deviation, and simple correlation coefficient). The results showed a significant correlation between (motor response speed) and forehand, smash shots. The results also showed a non-significant correlation between (motor response speed), long high serve, short low serve and backhand shot. The researcher recommends emphasizing training badminton players on special exercises to develop the motor response speed, and conducting similar research and studies dealing with the rest of the special physical characteristics of badminton and its association with other basic skills and according to its importance to the game.

The current research aims to: 1- Build an entrepreneurial trend in the halls of fitness and bodybuilding in the city of Mosul from the viewpoint of its visitors. 2- Investigate the level of leadership in the halls of fitness and body building in the city of Mosul from the viewpoint of its Visitors. The researchers used the descriptive approach in its survey to suit the nature of the research, and the research community included (378) people who went to the halls of fitness and body building in the city of Mosul, which were numbered (70) halls distributed on the right sides of the city with (10) halls, and the left with (60) Halls, thus, the research sample included (27) halls for fitness and body building in the city of Mosul, which represented (38.57%) of the research community, and these halls are divided into (9) halls on the right side and (18) halls on the left side of the city of Mosul. The number of people returning to these halls according to the scientific conditions of stability was (378) people distributed over the pilot sample with (18) people and the building sample with (260) people and (68.78%) from the research sample in addition to (100) people for the sample application. And by (26.45%) of the research sample, With regard to the research tools, the researcher built a scale of pioneering orientation for fitness halls and bodybuilding in the city of Mosul according to four dimensions (creativity, risk-taking, proactive, and competitive). On the final research sample in order to obtain data that were statistically treated through the percentage, the arithmetic mean, the simple correlation coefficient, and (T) test for two independent samples. The study concluded the following conclusions: - the level of entrepreneurship in the halls of fitness and bodybuilding in the city of Mosul ranged from medium to weak. The researchers recommended the following: 1- The departments of the halls of fitness and bodybuilding must adopt the pioneering approach in order to achieve the success for these halls. 2- The necessity for hall administrations to develop them by updating the devices and tools used as well as the other aspects in the hall.

Abstract In this article, we present the optimal bounds for a special elliptic integral mean in terms of the harmonic combinations of harmonic, geometric, and arithmetic means. As consequences, several new bounds for the complete elliptic integral of the second kind are discovered, which are the improvements of many previously known results.

This paper aims to develop the extension of intersection method for multi-objective optimization under condition of interval number. Based on the linear correlation of partial favourable probability and the corresponding performance indicator, and the assumption of uniform distribution of the actual value of performance indicator within the range of its lower and upper limits in case of interval number, it derives that the actual partial favourable probability of a performance indicator is the arithmetic mean value of the partial favourable probabilities of the arithmetic mean value and the variation value of the interval index of the corresponding performance indicator for each candidate, or their desired sum. Furthermore, according to the rule of algorithm for the total favourable probability quantitatively, all candidates are ranked according to their total favourable probabilities to complete the multi- objective optimization in case of interval number. As applications, the quantitative assessments of multi-criteria selections for effective dwelling house walls, project managers and contractor for construction works are given in detail, satisfied results are obtained.

In this paper, optimal bounds for the sine and hyperbolic tangent means by arithmetic and centroidal means in exponential type are established using the monotone form of L'Hospital's rule and the criterion for the monotonicity of the quotient of power series.

The goal of the studies described in the paper is to find a quantitative assessment that maximally correlates with the subjective assessment of the contrast image quality in the absence of reference image. As a result of the literature analysis, 16 functions were selected that are used for no-refernce image quality assessment: BEGH, BISH, BREN, CMO, CURV, FUS, HELM, EBCM, KURT, LAPD, LAPL, LAPM, LOCC, LOEN, SHAR, WAVS. They all use the arithmetical mean of the local contrast quality. As an alternative to averaging local estimates (since the mean is one of two parameters of the normal distribution), it is proposed to use one of two parameters of the Weibull distribution of the same data – scale or shape.For the experiments, digital images with nonlinear contrast distortion from the available CCID2014 database were used. It contains 15 original images with a size of 768x512 pixels and 655 versions with modified contrast. This database of images contains the average visual quality assessment (Mean Opinion Score, briefly MOS) of each image. Spearman’s rank correlation coefficient was used to determine the correspondence between the visual MOS scores and the studied quantitative measures.As a result of the research, a new quality assessment measure of contrast images in the absence of references is presented. To obtain the estimate, local quality values are calculated by the BREN measure, their set is described by the Weibull distribution, and the scale parameter of the distribution serves as the best numerical estimate of the quality of contrast images. This conclusion is confirmed experimentally, and the proposed measure correlates better than other variants with the subjective assessments of experts.

In the article, we present the sharp upper and lower bounds for the arithmetic mean in terms of new Seiffert-like means, which give some refinements of the results obtained in [1]. As applications, two new inequalities for the sine and hyperbolic sine functions will be established.

For $ a, b > 0 $ with $ a\neq b $, the Gauss lemniscate mean $ \mathcal{LM}(a, b) $ is defined by $ \begin{equation*} \mathcal{LM}(a,b) = \left\{\begin{array}{lll} \frac{\sqrt{a^2-b^2}}{\left[{ {\rm{arcsl}}}\left(\sqrt[4]{1-b^2/a^2}\right)\right]^2}, \ &a>b,\\ \frac{\sqrt{b^2-a^2}}{\left[{ {\rm{arcslh}}}\left(\sqrt[4]{b^2/a^2-1}\right)\right]^2},\ &a

Abstract In the paper, the authors define a new notion of “HT-convex function”, present some Hadamard-type inequalities for the new class of HT-convex functions and for the product of any two HT-convex functions, and derive some inequalities for the arithmetic mean and the p-logarithmic mean. These results generalize corresponding ones for HA-convex functions and MT-convex functions.

In this paper we obtain some new inequalities for Heinz operator mean.

Neurite orientation dispersion and density imaging (NODDI) estimates microstructural properties of brain tissue relating to the organisation and processing capacity of neurites, which are essential elements for neuronal communication. Descriptive statistics of NODDI tissue metrics are commonly analyzed in regions-of-interest (ROI) to identify brain-phenotype associations. Here, the conventional method to calculate the ROI mean weights all voxels equally. However, this produces biased estimates in the presence of CSF partial volume. This study introduces the tissue-weighted mean, which calculates the mean NODDI metric across the tissue within an ROI, utilising the tissue fraction estimate from NODDI to reduce estimation bias. We demonstrate the proposed mean in a study of white matter abnormalities in young onset Alzheimer's disease (YOAD). Results show the conventional mean induces significant bias that correlates with CSF partial volume, primarily affecting periventricular regions and more so in YOAD subjects than in healthy controls. Due to the differential extent of bias between healthy controls and YOAD subjects, the conventional mean under- or over-estimated the effect size for group differences in many ROIs. This demonstrates the importance of using the correct estimation procedure when inferring group differences in studies where the extent of CSF partial volume differs between groups. These findings are robust across different acquisition and processing conditions. Bias persists in ROIs at higher image resolution, as demonstrated using data obtained from the third phase of the Alzheimer's disease neuroimaging initiative (ADNI); and when performing ROI analysis in template space. This suggests that conventional ROI means of NODDI metrics are biased estimates under most contemporary experimental conditions, the correction of which requires the proposed tissue-weighted mean. The tissue-weighted mean produces accurate estimates of ROI means and group differences when ROIs contain voxels with CSF partial volume. In addition to NODDI, the technique can be applied to other multi-compartment models that account for CSF partial volume, such as the free water elimination method. We expect the technique to help generate new insights into normal and abnormal variation in tissue microstructure of regions typically confounded by CSF partial volume, such as those in individuals with larger ventricles due to atrophy associated with neurodegenerative disease.

The article is a follow-up study to research on analysing methodological approaches to estimate anthropogenic impacts on marine biota. The work examines relevant publications about the vulnerability of seabirds to the effects of offshore wind farms, relying upon the provisions of the measurement theory and the median theorem for ordinal quantities. It has been shown that the final ranking of sensitivity of certain bird species to wind farms, or indices of a risk of collisions with turbine blades and indices of a risk of birds’ displacement by wind farms, can vary at permissible monotonic transformations of the values of initial factors since all of these are estimated on ordinal scales. We conclude such estimates are incorrect. The summation of the two indices (exposure to collisions with turbine blades and to habitat change) of the birds’ species vulnerability in the context of the proposed models is incorrect. It has been demonstrated that the model based on dividing factors into primary and aggravation, when the latter are incorporated into the formula of computation through the exponent of primary factors, is incorrect: when primary factors have maximum values (equal to 1), the effect of aggravation factors is no longer taken into account; with some values of factors in the model, infinite vulnerability values can be obtained. The models are to operate within the entire presented range of initial data, but that is not fulfilled. Thus, none of the analysed models are correct, and an approach is needed, based on the use of metric values. Considering that the bird flight altitude primarily determines the impact of wind farms on birds, the proposals have been formulated to take into account the impact of various factors on the flight altitude.

Diffusion magnetic resonance imaging (MRI) is an increasingly popular technique in basic and clinical neuroscience. One promising application is to combine diffusion MRI with myelin maps from complementary MRI techniques such as multi-parameter mapping (MPM) to produce g-ratio maps that represent the relative myelination of axons and predict their conduction velocity. Statistical Parametric Mapping (SPM) can process both diffusion data and MPMs, making SPM the only widely accessible software that contains all the processing steps required to perform group analyses of g-ratio data in a common space. However, limitations have been identified in its method for reducing susceptibility-related distortion in diffusion data. More generally, susceptibility-related image distortion is often corrected by combining reverse phase-encoded images (blip-up and blip-down) using the arithmetic mean (AM), however, this can lead to blurred images. In this study we sought to (1) improve the susceptibility-related distortion correction for diffusion MRI data in SPM; (2) deploy an alternative approach to the AM to reduce image blurring in diffusion MRI data when combining blip-up and blip-down EPI data after susceptibility-related distortion correction; and (3) assess the benefits of these changes for g-ratio mapping. We found that the new processing pipeline, called consecutive Hyperelastic Susceptibility Artefact Correction (HySCO) improved distortion correction when compared to the standard approach in the ACID toolbox for SPM. Moreover, using a weighted average (WA) method to combine the distortion corrected data from each phase-encoding polarity achieved greater overlap of diffusion and more anatomically faithful structural white matter probability maps derived from minimally distorted multi-parameter maps as compared to the AM. Third, we showed that the consecutive HySCO WA performed better than the AM method when combined with multi-parameter maps to perform g-ratio mapping. These improvements mean that researchers can conveniently access a wide range of diffusion-related analysis methods within one framework because they are now available within the open-source ACID toolbox as part of SPM, which can be easily combined with other SPM toolboxes, such as the hMRI toolbox, to facilitate computation of myelin biomarkers that are necessary for g-ratio mapping.

Abstract In the article, we provide the sharp bounds for the Sándor–Yang mean in terms of certain families of the two-parameter geometric and arithmetic mean and the one-parameter geometric and harmonic means. As applications, we present new bounds for a certain Yang mean and the inverse tangent function.

Abstract In the article, we prove that λ1=1/2+[(2+log(1+2))/2]1/ν−1/2 $\lambda _{1}=1/2+\sqrt{ [ (\sqrt{2}+ \log (1+\sqrt{2}) )/2 ]^{1/\nu }-1}/2$, μ1=1/2+6ν/(12ν) $\mu _{1}=1/2+\sqrt{6 \nu }/(12\nu )$, λ2=1/2+[(π+2)/4]1/ν−1/2 $\lambda _{2}=1/2+\sqrt{ [(\pi +2)/4 ] ^{1/\nu }-1}/2$ and μ2=1/2+3ν/(6ν) $\mu _{2}=1/2+\sqrt{3\nu }/(6\nu )$ 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 $x\neq y$ and ν∈[1/2,∞) $\nu \in [1/2, \infty )$, 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) $\mathcal{R}_{QA}(x, y)$ and RAQ(x,y) $\mathcal{R}_{AQ}(x, y)$ are two Neuman means.

Abstract In the article, we provide new bounds for two Sándor–Yang means in terms of the arithmetic and contraharmonic means. Our results are the improvements of the previously known results.

Abstract 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 $a\neq b$ if and only if α1≤3/16=0.1875 $\alpha_{1}\leq 3/16=0.1875$, β1≥64/π2−6=0.484555… $\beta_{1}\geq64/\pi^{2}-6= 0.484555\dots$, α2≤3/16=0.1875 $\alpha_{2}\leq3/16=0.1875$ and β2≥(5log2−log3−2logπ)/(log7−log6)=0.503817… $\beta_{2}\geq(5\log2-\log3-2\log \pi)/(\log7-\log6)= 0.503817\dots$, where E(a,b)=(2π∫0π/2acos2θ+bsin2θdθ)2 $E(a,b)= (\frac{2}{\pi}\int^{\pi/2}_{0}\sqrt{a\cos^{2}\theta +b\sin^{2}\theta}\,d\theta )^{2}$, H(a,b)=2ab/(a+b) $H(a,b)=2ab/(a+b)$, G(a,b)=ab $G(a,b)=\sqrt{ab}$, 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 this paper, we establish new generalizations of the Hermite-Hadamard-type inequalities. These inequalities are formulated in terms of modules of certain powers of proper functions. Generalizations for convex functions are also considered. As applications, some new inequalities for the digamma function in terms of the trigamma function and some inequalities involving special means of real numbers are given. The results also include estimates via arithmetic, geometric and logarithmic means. The examples are derived in order to demonstrate that some of our results in this paper are more exact than the existing ones and some improve several known results available in the literature. The constants in the derived inequalities are calculated; some of these constants are sharp. As a visual example, graphs of some technically important functions are included in the text.

In this paper, the authors provide several sharp upper and lower bounds for the Neuman–Sándor mean in terms of the arithmetic and contra-harmonic means, and present some new sharp inequalities involving hyperbolic sine function and hyperbolic cosine function.

By using the power series of the functions 1/sinnt and cost/sinnt (n=1,2,3,4,5), and the estimation of the ratio of two adjacent Bernoulli numbers, we obtained new bounds for arithmetic mean A by the weighted arithmetic means of Mtan1/3Msin2/3 and 13Mtan+23Msin,Mtanh1/3Msinh2/3 and 13Mtanh+23Msinh, where Mtan(x,y) and Msin(x,y), Mtanh(x,y) and Msinh(x,y) are the tangent mean, sine mean, hyperbolic tangent mean and hyperbolic sine mean, respectively. The upper and lower bounds obtained in this paper are compared in detail with the conclusions of the previous literature.

Abstract For t∈[0,1/2] $t\in [0,1/2]$ and s≥1 $s\ge 1$, we consider the two-parameter family of means Qt,s(a,b)=Gs(ta+(1−t)b,(1−t)a+tb)A1−s(a,b), $$ Q_{t,s}(a,b)=G^{s}\bigl(ta+(1-t)b,(1-t)a+tb\bigr)A^{1-s}(a,b), $$ where A and G denote the arithmetic and geometric means. Sharp bounds for the identric mean in terms of Qt,s $Q_{t,s}$ are obtained. Our results generalize and extend bounds due to Chu et al. (Abstr. Appl. Anal. 2011:657935, 2011) and to Wang et al. (Appl. Math. Lett. 25:471–475, 2012).

Abstract In the article, we provide several sharp upper and lower bounds for two Sándor–Yang means in terms of combinations of arithmetic and contra-harmonic means.

Abstract In the article, we establish several Ostrowski type inequalities involving the conformable fractional integrals. As applications, we find new inequalities for the arithmetic and generalized logarithmic means.

Refining and reversing weighted arithmetic-geometric mean inequalities have been studied in many papers. In this paper, we provide some bounds for the differences between the weighted arithmetic and geometric means, using known inequalities. We improve the results given by Furuichi-Ghaemi-Gharakhanlu and Sababheh-Choi. We also give some bounds on entropies, applying the results in a different approach. We explore certain convex or concave functions, which are symmetric functions on the axis t=1/2.

Despite the numerous papers on the statistical analyses of pH, there is no explicit opinion on the use of arithmetic mean as a measure of the central tendency for pH and H+ activity. The problem arises because the transformation of the arithmetic mean for one does not give the arithmetic mean for the other. The paper presents 1) the theoretical considerations on the distribution of pH and H+ activity and relation between them, properties of these distributions, the choice of distributions which should be consistent with the distribution of pH and the distribution of H+ activity and measures of central tendency for features of such distributions and 2) examples of calculations of measures of central tendency for pH and H+ activity based on the literature data on soil and lake water pH. These data analyses included distributions of pH and H+ activities, properties of distribution, descriptive statistics for pH and for the H+ activity and comparison of arithmetic mean with the geometric mean. From the results, it could be concluded that a uniform approach to the choice of measure for the central tendency of pH and H+ activity requires the determination of the type of measure (mean) for one of them and then consistent transformation of this measure. The choice of measure of the central tendency for the variable should be preceded by determination of its distribution. Normal probability distribution of pH and thus lognormal distribution of H+ activity indicate that the arithmetic mean, and its corresponding geometric mean should be used as proper measures of the central tendency for pH and for H+ activity. Besides, the position statistic that is a median can be used for each of those variables, irrespective of their probability distributions.

Abstract In the article, we present the best possible parameters λ = λ ( p ) $\lambda=\lambda (p)$ and μ = μ ( p ) $\mu=\mu(p)$ on the interval [ 0 , 1 / 2 ] $[0, 1/2]$ 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 ) $$\begin{aligned}& G^{p}\bigl[\lambda a+(1-\lambda)b, \lambda b+(1-\lambda)a \bigr]A^{1-p}(a,b) \\& \quad< E(a,b) < G^{p}\bigl[\mu a+(1-\mu)b, \mu b+(1-\mu)a \bigr]A^{1-p}(a,b) \end{aligned}$$ holds for any p ∈ [ 1 , ∞ ) $p\in[1, \infty)$ and all a , b > 0 $a, b>0$ with a ≠ b $a\neq b$ , where A ( a , b ) = ( a + b ) / 2 $A(a, b)=(a+b)/2$ , G ( a , b ) = a b $G(a,b)=\sqrt{ab}$ and E ( a , b ) = [ 2 ∫ 0 π / 2 a cos 2 θ + b sin 2 θ d θ / π ] 2 $E(a,b)=[2\int_{0}^{\pi /2}\sqrt{a\cos^{2}\theta+b\sin^{2}\theta}\,d\theta/\pi]^{2}$ are the arithmetic, geometric and special quasi-arithmetic means of a and b, respectively.

The aim of the paper is to present some mean value theorems obtained as consequences of the intermediate value property. First, we will prove that any nonextremum value of a Darboux function can be represented as an arithmetic, geometric or harmonic mean of some different values of this function. Then, we will present some extensions of the Cauchy or Lagrange Theorem in classical or integral form. Also, we include similar results involving divided differences. The paper was motivated by some problems published in mathematical journals.

Abstract In this paper, we present the best possible parameters α ( r ) $\alpha(r)$ and β ( r ) $\beta(r)$ such that the double inequality [ α ( r ) A r ( a , b ) + ( 1 − α ( r ) ) Q r ( a , b ) ] 1 / r < T D [ A ( a , b ) , Q ( a , b ) ] < [ β ( r ) A r ( a , b ) + ( 1 − β ( r ) ) Q r ( a , b ) ] 1 / r $$\begin{aligned} \bigl[\alpha(r)A^{r}(a,b)+ \bigl(1-\alpha(r) \bigr)Q^{r}(a,b) \bigr]^{1/r} < & TD \bigl[A(a,b), Q(a,b) \bigr] \\ < & \bigl[\beta(r)A^{r}(a,b)+ \bigl(1-\beta(r) \bigr)Q^{r}(a,b) \bigr]^{1/r} \end{aligned}$$ holds for all r ≤ 1 $r\leq 1$ and a , b > 0 $a, b>0$ with a ≠ b $a\neq b$ , and we provide new bounds for the complete elliptic integral E ( r ) = ∫ 0 π / 2 ( 1 − r 2 sin 2 θ ) 1 / 2 d θ $\mathcal{E}(r)=\int_{0}^{\pi/2}(1-r^{2}\sin^{2}\theta)^{1/2}\,d\theta$ ( r ∈ ( 0 , 2 / 2 ) ) $(r\in (0, \sqrt{2}/2))$ of the second kind, where T D ( a , b ) = 2 π ∫ 0 π / 2 a 2 cos 2 θ + b 2 sin 2 θ d θ $TD(a,b)=\frac{2}{\pi}\int_{0}^{\pi/2}\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}\,d\theta$ , A ( a , b ) = ( a + b ) / 2 $A(a,b)=(a+b)/2$ and Q ( a , b ) = ( a 2 + b 2 ) / 2 $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$ are the Toader, arithmetic, and quadratic means of a and b, respectively.

In this paper we study a certain differential subordination related to the harmonic mean and its symmetry properties, in the case where a dominant is a linear function. In addition to the known general results for the differential subordinations of the harmonic mean in which the dominant was any convex function, one can study such differential subordinations for the selected convex function. In this case, a reasonable and difficult issue is to look for the best dominant or one that is close to it. This paper is devoted to this issue, in which the dominant is a linear function, and the differential subordination of the harmonic mean is a generalization of the Briot–Bouquet differential subordination.

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.

We give bounds on the difference between the weighted arithmetic mean and the weighted geometric mean. These imply refined Young inequalities and the reverses of the Young inequality. We also studied some properties on the difference between the weighted arithmetic mean and the weighted geometric mean. Applying the newly obtained inequalities, we show some results on the Tsallis divergence, the Rényi divergence, the Jeffreys–Tsallis divergence and the Jensen–Shannon–Tsallis divergence.

Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the "power triangle inequality". The power triangle inequality represents an uncountably large class of inequalities, and includes the triangle inequality, relaxed triangle inequality, and inframetric inequality as special cases. The power triangle inequality is defined in terms of a function that is herein called the power triangle function. The power triangle function is itself a power mean, and as such is continuous and monotone with respect to its exponential parameter, and also includes the operations of maximum, minimum, mean square, arithmetic mean, geometric mean, and harmonic mean as special cases.

Abstract In the article, we prove that the double inequalities M α ( a , b ) < S Q A ( a , b ) < M β ( a , b ) $M_{\alpha }(a,b)< S_{QA}(a,b)< M_{\beta}(a,b)$ and M λ ( a , b ) < S A Q ( a , b ) < M μ ( a , b ) $M_{\lambda }(a,b)< S_{AQ}(a,b)< M_{\mu}(a,b)$ hold for all a , b > 0 $a, b>0$ with a ≠ b $a\neq b$ if and only if α ≤ log 2 / [ 1 + log 2 − 2 log ( 1 + 2 ) ] = 1.5517 … $\alpha\leq\log 2/[1+\log2-\sqrt{2}\log(1+\sqrt{2})]=1.5517\ldots$ , β ≥ 5 / 3 $\beta\geq5/3$ , λ ≤ 4 log 2 / [ 4 + 2 log 2 − π ] = 1.2351 … $\lambda\leq4\log2/[4+2\log2-\pi]=1.2351\ldots$ and μ ≥ 4 / 3 $\mu\geq4/3$ , where S Q A ( a , b ) = A ( a , b ) e Q ( a , b ) / M ( a , b ) − 1 $S_{QA}(a,b)=A(a,b)e^{Q(a,b)/M(a,b)-1}$ and S A Q ( a , b ) = Q ( a , b ) e A ( a , b ) / T ( a , b ) − 1 $S_{AQ}(a,b)=Q(a,b)e^{A(a,b)/T(a,b)-1}$ are the Sándor-type means, A ( a , b ) = ( a + b ) / 2 $A(a,b)=(a+b)/2$ , Q ( a , b ) = ( a 2 + b 2 ) / 2 $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$ , T ( a , b ) = ( a − b ) / [ 2 arctan ( ( a − b ) / ( a + b ) ) ] $T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$ , and M ( a , b ) = ( a − b ) / [ 2 sinh − 1 ( ( a − b ) / ( a + b ) ) ] $M(a,b)=(a-b)/[2\sinh ^{-1}((a-b)/(a+b))]$ are, respectively, the arithmetic, quadratic, second Seiffert, and Neuman-Sándor means.

Abstract In this paper, we present the best possible parameters p and q such that the double inequality M p ( a , b ) < V ( a , b ) < M q ( a , b ) $$ M_{p}(a,b)< V(a,b)< M_{q}(a,b) $$ holds for all a , b > 0 $a, b>0$ with a ≠ b $a\neq b$ , where M r ( a , b ) = [ ( a r + b r ) / 2 ] 1 / r $M_{r}(a,b)=[(a^{r}+b^{r})/2]^{1/r}$ ( r ≠ 0 $r\neq0$ ) and M 0 ( a , b ) = a b $M_{0}(a,b)= \sqrt {ab}$ is the rth power mean and V ( a , b ) = ( a − b ) / [ 2 sinh − 1 ( ( a − b ) / 2 a b ) ] $V(a,b)=(a-b)/[\sqrt{2}\sinh^{-1}((a-b)/\sqrt{2ab})]$ is the second Yang mean.

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

We propose a genetic algorithm for a weighted p-median problem. It is a facility location problem. The algorithm generates a good solution quickly. Computational tests were realized on five different tasks from 21 vertices to 100 vertices and from p-median from p=3 to p=6. The tests were performed 100 times on every task. There were created some modifications of these tasks for a proposed genetic algorithm. The best solution generated by this algorithm is within 0.6% of the optimum for 80% of the tasks. The other 20% of the tasks is within 1.6% of the optimum. Time of realization is within 5.9 s.

The most frequently used instrument for measuring velocity distribution in the cross-section of small rivers is the propeller-type current meter. Output of measuring using this instrument is point data of a tiny bulk. Spatial interpolation of measured data should produce a dense velocity profile, which is not available from the measuring itself. This paper describes the preparation of interpolation models.

This paper addresses the problem of estimating the finite population variance using auxiliary information in sample surveys. Motivated by (Singh and Vishwakarma, 2009) some estimators of finite population variance have been suggested along with their properties in simple random sampling. The theoretical conditions under which the proposed estimators are more efficient than usual unbiased, usual ratio and (Singh et al., 2009) estimators have been obtained. Numerical illustrations are given in support of the present study.

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.

Statistics today represent a group of scientific methods for the quantitative and qualitative investigation of variations in mass appearances. In fact, statistics present a group of methods that are used for the accumulation, analysis, presentation and interpretation of data necessary for reaching certain conclusions. Statistical analysis is divided into descriptive statistical analysis and inferential statistics. The values which represent the results of an experiment, and which are the subject of observation of a certain occurrence, are called parameters and they are divided into descriptive and numerical. All numerical parameters are divided into non-continuous and continuous. The graphic presentation of the distribution of frequencies can be by poligon or histogram. The most frequently applied descriptive statistical methods are: arithmetic mean, standard deviation, standard error of arithmetic mean, variation coefficient, and variation interval.

The common practice of counting bacteria using epifluorescence microscopy involves selecting 5–30 random fields of view on a glass slide to calculate the arithmetic mean which is then used to estimate the total bacterial abundance. However, not much is known about the accuracy of the arithmetic mean when it is calculated by selecting random fields of view and its effect on the overall abundance. The aim of this study is to evaluate the accuracy and reliability of the arithmetic mean by estimating total bacterial abundance and to calculate its variance using a bootstrapping technique. Three fixed suspensions obtained from a three-week-old marine biofilm were stained and dispersed on glass slides. Bacterial cells were counted from a total of 13,924 fields of view on each slide. Total bacterial count data obtained were used for calculating the arithmetic mean and associated variance and bias for sample field sizes of 5, 10, 15, 20, 25, 30, 35 and 40. The study revealed a non-uniform distribution of bacterial cells on the glass slide. A minimum of 20 random fields of view or a minimum of 350 bacterial cells need to be counted to obtain a reliable value of the arithmetic mean to estimate the total bacterial abundance for a marine biofilm sample dispersed on a glass slide.

In this paper, we find the greatest values \(\alpha\) and \(\lambda\), and the least values \(\beta\) and \(\mu\) such that the double inequalities \[C^{\alpha}(a,b)A^{1-\alpha}(a,b)0\) with \(a\neq b\), where \(M(a,b)\), \(A(a,b)\) and \(C(a,b)\) denote the Neuman-Sándor, arithmetic, and contra-harmonic means of \(a\) and \(b\), respectively.

In this paper, we find the greatest values \(\alpha\) and \(\lambda\), and the least values \(\beta\) and \(\mu\) such that the double inequalities \[C^{\alpha}(a,b)A^{1-\alpha}(a,b)0\) with \(a\neq b\), where \(M(a,b)\), \(A(a,b)\) and \(C(a,b)\) denote the Neuman-Sándor, arithmetic, and contra-harmonic means of \(a\) and \(b\), respectively.

For \(p\in \mathbb{R}\), the power mean \(M_{p}(a,b)\) of order \(p\), identric mean \(I(a,b)\) and arithmetic mean \(A(a,b)\) of two positive real numbers \(a\) and \(b\) are defined by \begin{equation*} M_{p}(a,b)= \begin{cases} \displaystyle\left(\tfrac{a^{p}+b^{p}}{2}\right)^{1/p}, & p\neq 0,\\ \sqrt{ab}, & p=0, \end{cases} \quad I(a,b)= \begin{cases} \displaystyle\tfrac{1}{\rm {e}}\left(b^{b}/a^{a}\right)^{1/(b-a)}, & a\neq b,\\ \displaystyle a, & a=b, \end{cases} \end{equation*} and \(A(a,b)=(a+b)/2\), respectively. In the article, we answer the questions: What are the least values \(p\), \(q\) and \(r\), such that inequalities \(A^{1/2}(a,b)I^{1/2}(a,b)\leq M_{p}(a,b)\), \(A(a,b)^{1/3}I^{2/3}(a,b)\leq M_{q}(a,b)\) and \(A^{2/3}(a,b)I^{1/3}(a,b)\leq M_{r}(a,b)\) hold for all \(a,b>0\)?

For \(p\in \mathbb{R}\), the power mean \(M_{p}(a,b)\) of order \(p\), identric mean \(I(a,b)\) and arithmetic mean \(A(a,b)\) of two positive real numbers \(a\) and \(b\) are defined by \begin{equation*} M_{p}(a,b)= \begin{cases} \displaystyle\left(\tfrac{a^{p}+b^{p}}{2}\right)^{1/p}, & p\neq 0,\\ \sqrt{ab}, & p=0, \end{cases} \quad I(a,b)= \begin{cases} \displaystyle\tfrac{1}{\rm {e}}\left(b^{b}/a^{a}\right)^{1/(b-a)}, & a\neq b,\\ \displaystyle a, & a=b, \end{cases} \end{equation*} and \(A(a,b)=(a+b)/2\), respectively. In the article, we answer the questions: What are the least values \(p\), \(q\) and \(r\), such that inequalities \(A^{1/2}(a,b)I^{1/2}(a,b)\leq M_{p}(a,b)\), \(A(a,b)^{1/3}I^{2/3}(a,b)\leq M_{q}(a,b)\) and \(A^{2/3}(a,b)I^{1/3}(a,b)\leq M_{r}(a,b)\) hold for all \(a,b>0\)?

For \(p\in\mathbb{R}\), the one-parameter mean \(J_{p}(a,b)\), arithmetic mean \(A(a,b)\), and harmonic mean \(H(a,b)\) of two positive real numbers \(a\) and \(b\) are defined by\begin{equation*}J_{p}(a,b)=\begin{cases}\tfrac{p(a^{p+1}-b^{p+1})}{(p+1)(a^p-b^p)}, & a\neq b,p\neq 0,-1,\\\tfrac{a-b}{\log{a}-\log{b}}, & a\neq b,p=0,\\\tfrac{ab(\log{a}-\log{b})}{a-b}, & a\neq b,p=-1,\\a, & a=b,\end{cases}\end{equation*}\(A(a,b)=\tfrac{a+b}{2}\), and \(H(a,b)=\tfrac{2ab}{a+b}\),respectively. In this paper, we answer the question: For \(\alpha\in(0,1)\), what are the greatest value \(r_{1}\) and the least value \(r_{2}\) such that the double inequality \(J_{r_{1}}(a,b)0\) with \(a\neq b\)?

The logarithmic mean \(L(a,b)\), identric mean \(I(a,b)\), arithmeticmean \(A(a,b)\) and harmonic mean \(H(a,b)\) of two positive real values \(a\) and \(b\) are defined by\begin{align*}\label{main}&L(a,b)=\begin{cases}\tfrac{b-a}{\log b-\log a},& a\neq b,\\a,&a=b,\end{cases}\\&I(a,b)=\begin{cases}\tfrac{1}{{\rm e}}\left(\tfrac{b^b}{a^a}\right)^{\tfrac{1}{b-a}},& a\neq b,\\a,&a=b,\end{cases}\end{align*}\(A(a,b)=\tfrac{a+b}{2}\) and \(H(a,b)=\tfrac{2ab}{a+b}\), respectively. In this article, we answer the questions: What are the best possible parameters \(\alpha_{1},\alpha_{2},\beta_{1}\) and \(\beta_{2}\), such that \(\alpha_{1}A(a,b)+(1-\alpha_{1})H(a,b)\leq L(a,b)\leq\beta_{1}A(a,b)+(1-\beta_{1})H(a,b)\) and \(\alpha_{2}A(a,b)+(1-\alpha_{2})H(a,b)\leq I(a,b)\leq\beta_2A(a,b)+(1-\beta_{2})H(a,b)\) hold for all \(a,b>0\)?