Use Of Geometric Mean Research Articles

The Upgraded Planck System of Units That Reaches from the Known Planck Scale All the Way down to Subatomic Scales

Natural systems of units {Ui} need to be overhauled to include the dimensionless coupling constants {αUi} of the natural forces. Otherwise, they cannot quantify all the forces of nature in a unified manner. Thus, each force must furnish a system of units with at least one dimensional and one dimensionless constant. We revisit three natural systems of units (atomic, cosmological, and Planck). The Planck system is easier to rectify, and we do so in this work. The atomic system discounts {G,αG}, thus it cannot account for gravitation. The cosmological system discounts {h,αh}, thus it cannot account for quantum physics. Here, the symbols have their usual meanings; in particular, αG is the gravitational coupling constant and αh is Dirac’s fine-structure constant. The speed of light c and the impedance of free space Z0 are resistive properties imposed by the vacuum itself; thus, they must be present in all systems of units. The upgraded Planck system with fundamental units UPS:={c,Z0,G,αG,h,αh,…} describes all physical scales in the universe—it is nature’s system of units. As such, it reveals a number of properties, most of which have been encountered previously in seemingly disjoint parts of physics and some of which have been designated as mere coincidences. Based on the UPS results, which relate (sub)atomic scales to the Planck scale and the fine-structure constant to the Higgs field, we can state with confidence that no observed or measured physical properties are coincidental in this universe. Furthermore, we derive from first principles Koide’s K=2/3 enigmatic constant and additional analogous quark and vector boson constants. These are formal mathematical proofs that justify a posteriori the use of geometric means in deriving the quark/boson mass ladder. This ladder allows us to also calculate the Higgs couplings to the vector bosons and the Weinberg angle in terms of K only, and many of the “free” parameters of the Standard Model of particle physics were previously expected to be determined only from experiments.

Formation of students' mental activity by improving the applied focus of learning a geometric

Prerequisites for the study the problems of implementing the applied direction of teaching the topic of geometric construction in the geometry course of secondary school are most relevant, since there are no clear basic methods for the formation of students' thinking activities in this direction. This article discusses the results of a scientific study aimed at obtaining theoretical knowledge in any section of school geometry, which aims to give the student the skills to use the means of drawing the properties of geometric shapes and relationships. For this purpose, students were given a review of the literature on the implementation of the applied direction of geometry, focusing on mathematical statements in order to improve the applied direction of geometry. In the framework of the topic, the analysis was carried out in the course of improving this topic, due to the fact that mathematics teachers did not pay enough attention to the problems of construction in the geometry course of secondary school due to lack of time, students did not develop skills in solving problems of the applied direction of geometry. As a result of research, geometric construction problems are an important element of the mathematical preparation of schoolchildren. Using the problems of geometric construction, its application points were determined. The ways of implementing the applied direction in teaching geometry to students are described. Also, when teaching the geometry course, problems of the applied direction were collected in the school's geometry textbooks, ways to solve them were considered, and plot geometric tasks were prepared in order to increase the creativity of students. The stages of solving the practice-oriented geometric construction problem and the principles of implementing the applied direction of teaching mathematics, defined as a special type of plot problem, reflecting the use of geometric means in practical situations, are also studied. In the future, it can be used as a methodological tool for mathematics teachers in teaching construction problems.

IGFBP-7 and Outcomes in Heart Failure With Reduced Ejection Fraction: Findings From DAPA-HF

BackgroundInsulin-like growth factor–binding protein-7 (IGFBP-7) has been proposed as a potential prognostic biomarker in heart failure (HF), but the association between elevation in IGFBP-7 and HF outcomes in ambulant patients with heart failure with reduced ejection fraction (HFrEF) is unknown. ObjectivesThe authors addressed this question in a post hoc analysis of the DAPA-HF (Dapagliflozin and Prevention of Adverse Outcomes in Heart Failure) trial. MethodsThe primary outcome was a composite of cardiovascular death or a worsening HF event. The risk of adverse outcome was compared across tertiles of IGFBP-7 concentration by means of Cox proportional hazard models adjusted for N-terminal pro–B-type natriuretic peptide (NT-proBNP) and high-sensitivity troponin T (hsTnT). The efficacy of randomized treatment across IGFBP-7 tertiles was assessed. Change in IGFBP-7 at 12 months was compared with the use of geometric means. ResultsA total of 3,158 patients had IGFBP-7 measured at baseline, and 2,493 had a repeated measure at 12 months. Patients in the highest tertile of IGFBP-7 had evidence of more advanced HFrEF. The adjusted HR for the primary endpoint in tertile 3, compared with tertile 1, was 1.48 (95% CI: 1.17-1.88). There was no modification of the benefit of dapagliflozin by baseline IGFBP-7 (P interaction = 0.34). Dapagliflozin did not change IGFBP-7 levels over 1 year (P = 0.34). ConclusionsHigher IGFBP-7 in patients with HFrEF was associated with worse clinical profile and an increased risk of adverse clinical outcomes. IGFBP-7 provided prognostic information incremental to clinical variables, NT-proBNP, and hsTnT. The benefit of dapagliflozin was not modulated by IGFBP-7 level. (Study to Evaluate the Effect of Dapagliflozin on the Incidence of Worsening Heart Failure or Cardiovascular Death in Patients With Chronic Heart Failure [DAPA-HF]; NCT03036124)

The impact of class imbalance in classification performance metrics based on the binary confusion matrix

A major issue in the classification of class imbalanced datasets involves the determination of the most suitable performance metrics to be used. In previous work using several examples, it has been shown that imbalance can exert a major impact on the value and meaning of accuracy and on certain other well-known performance metrics. In this paper, our approach goes beyond simply studying case studies and develops a systematic analysis of this impact by simulating the results obtained using binary classifiers. A set of functions and numerical indicators are attained which enables the comparison of the behaviour of several performance metrics based on the binary confusion matrix when they are faced with imbalanced datasets. Throughout the paper, a new way to measure the imbalance is defined which surpasses the Imbalance Ratio used in previous studies. From the simulation results, several clusters of performance metrics have been identified that involve the use of Geometric Mean or Bookmaker Informedness as the best null-biased metrics if their focus on classification successes (dismissing the errors) presents no limitation for the specific application where they are used. However, if classification errors must also be considered, then the Matthews Correlation Coefficient arises as the best choice. Finally, a set of null-biased multi-perspective Class Balance Metrics is proposed which extends the concept of Class Balance Accuracy to other performance metrics.

Bioconcentration factors and the risk concentrations of potentially toxic elements in garden soils

Empirical models describe soil-plant transfers to explain the variations in the occurrence of potentially toxic elements (PTE) in soils and to estimate the Bioconcentration Factor (BCF). In this study, results were selected based on data in the literature on soils of humid tropical and temperate regions to evaluate soil-plant transfer models, to calculate the BCF and to derive risk concentrations of Cu, Cr, Pb, Ni and Zn present in the exposure pathway leading to the consumption of contaminated vegetables. The Cetesb (Environmental Agency of the State of Sao Paulo, Brazil) mathematical model was used to derive the risk posed by soil concentrations in urban and rural exposure scenarios. The results of the pseudo total contents of PTE in the soil and the contents absorbed by plants were compared and the BCFs were calculated by the use of geometric means, including a correction factor appropriate to each particular type of soil. Differences were observed between BCFs calculated for each climate region: humid tropical (HTR) and temperate (TE), which the first one presented the highest values to BCF in leaves and the lowest BCF values for root, except Ni, compared to second one. The soil concentrations with the highest risk were found in humid tropical regions as compared with those found in temperate regions, except for Ni. The obtained BCFs may contribute to any future revisions of guideline values as well as help other state environmental agencies to establish their own guideline values.

Use of Geometric Mean in Bioequivalence Trials

Bioequivalence data often do not follow the normality assumption on the linear (original) scale, therefore in that situation, the use of the logarithmic transformation is recommended. In the bioequivalence analysis, confusion arises about the use of geometric mean ratio when the logarithmic transformation is recommended by the regulatory authorities. The purpose of this research paper is to clear this confusion. Different average bioequivalence criteria are also reviewed in this paper.

An Investigation of the Variability of Particulate Emissions from Woodstoves in New Zealand

Woodburning is the dominant source of PM10 in most of New Zealand, and the principal cause of exceedances of the national air quality 24-hr standard (50 µg m–3). Over the last decade New Zealand researchers have used multiple techniques to characterise woodstove emissions, including seven campaigns to measure real-life PM10 emissions from in-situ woodstoves. All measurements have shown considerable variation, both within and between campaigns. PM10 emission factors from in-situ tests exhibit a log-normal distribution with a geometric mean (± standard deviation) of 9.8 g kg–1 (± 2.4 g kg–1) and 3.9 g kg–1 (± 3.8 g kg–1) (dry wood) for older and low-emission stoves respectively. Since the distribution is log-normal we recommend the use of geometric mean rather than arithmetic mean as the emissions factor used for inventories. This paper examines the variability of PM10 emissions from woodstoves using a Kernel Density Estimate to compare the distributions within and between campaigns and correlations between coefficients of variations (c.v.) to see whether the variation in emissions factors could be reduced by the collection of further measurements to increase the size of the total dataset. We conclude that variation in emissions is inherent in the way woodstoves are used in real-life and that regulators will need to allow for a range of emissions in management plans as a definitive emission factor may not be possible.

Nonconcavity of mass exponents’ spectrum in multifractal analysis of community spatial structure: The problem and possible solutions

Species–area relationship and species abundance distribution are two of the most studied patterns in community ecology. Multifractal analysis, combining elements of analysis of both patterns in a unified framework, is a very promising method for studying the community spatial structure. However, most empirical applications of multifractal analysis revealed an abnormal result: an increasing spectrum of Renyi generalised dimensions. We argue that an abnormal form of another function, namely, mass exponents’ spectrum, is a more important problem. We hypothesise that the reason for the abnormal results is the spatial heterogeneity of the species abundance distribution. This leads to disproportionate contribution of subplots with high dominance or extreme rarity during averaging over the whole area. The effect arises from the use of arithmetic mean. We propose two new algorithms for calculation of mass exponents to solve the specified problem based, first, on the use of geometric mean and, second, on the calculation of local exponents and their subsequent averaging. We test the proposed approaches on the material of multifractal analysis of the spatial structure of the two communities: tropical moist forest and mixed deciduous forest. Results show that both communities are characterised by scale-specific heterogeneity of relative abundance extrema. New algorithms produce correct Renyi spectra, but mass exponents’ spectra do not get strict concavity in all cases, although the range of concavity expands. As a rule of thumb, we propose to use truncated multifractal spectra constructed from the mass exponents in the range of strict concavity. The use of truncated spectra allows obtaining biologically interpretable information about community structure, which enables ecological analysis to advance much further than with the use of Renyi spectra.

Land-use induced spatial heterogeneity of soil hydraulic properties on the Loess Plateau in China

On the Loess Plateau in China, soil erosion amounts to between 10 000 and 25 000 tons/km 2/year. In 1998, the EROCHINA project was started, with the major objective of developing alternative land-use and soil and water conservation strategies for the Loess Plateau, using the LISEM soil erosion model. In order to provide the model with accurate input on soil hydraulic properties of the catchment, this study tried to quantify these properties for major land-use units and to examine the effects of the statistically identified in-field heterogeneity on model outcome. The study area (Danangou catchment) is located in the middle part of the Loess Plateau in the northern part of Shaanxi Province. The catchment is about 3.5 km 2 in size. To determine the hydraulic properties of the soil, a sampling scheme was implemented to measure unsaturated conductivity and water retention characteristics. The saturated conductivity measurements were performed on land-use clusters, based on treatment and plant and soil differences. A 100×100 m sampling grid was set out on 12 fields, with 1×1 m grid squares. On each field, 10 sampling spots were randomly selected, using Simple Random Sampling. The sensitivity of the LISEM model to the measured heterogeneity of saturated conductivity was analysed by using the geometric mean and standard deviation (S.D.) of the K s-optimized of the various land-use units to calculate discharge and soil loss during a single rain event. This proved that, using the standard deviation of the saturated conductivity, which was 30–50%, the calculated discharge and total sediment losses varied by a factor 2. Using the standard deviation had a minor effect on the calculated sediment concentration. As regards the on-site effects, the use of the geometric mean of K s minus the S.D. resulted in an increase in the level of erosion, while the use of geometric mean of K s plus the S.D. value resulted in a significant decrease in erosion level relative to that obtained using the geometric mean of K s itself. Using randomly selected sampling spots and a calibration procedure as was done in the present study make detailed information on K s available, which can be used to compare alternative land-use options for their effectiveness in reducing discharge and sediment losses.

Growing food crops on sludge‐amended soils: Problems with the U.S. Environmental Protection Agency method of estimating toxic metal transfer

AbstractThe use of sewage sludges as farm fertilizers, encouraged in recent years by changes in U.S. Environmental Protection Agency (U.S. EPA) policy, has raised concerns among some scientists regarding food safety and long‐term soil productivity. The U.S. EPA risk assessment for entry into the human diet of three of the most toxic metals, cadmium (Cd), mercury (Hg), and lead (Pb), utilized uptake coefficients (UCs) to calculate the amount of each metal that could enter food crops from the soil. Each UC was calculated as the increment of metal concentration in the edible part of the crop per unit increase of metal loading to the soil. However, the final UC estimates employed in the risk assessment are biased toward low values by a number of factors. These include the use of geometric means to obtain single‐point averages of UCs for each crop group evaluated, rather than using arithmetic means or probabilistic methods, a systematic analytical or contamination error apparent in the reported metal concentrations of the control crops, and the fact that most of the UC values were derived for soils with pH 6 or higher. For more than 50% of all the soil and cropping conditions represented in the risk assessment, the geometrically averaged Cd UC values used by the U.S. EPA underestimated the actual risk posed by uptake into crops. The UC values for Pb and Hg are uncertain because of analytical or contamination errors, and because of the few data available for a number of crops. These uncertainties and biases in the risk assessment would advise a more cautious approach to agricultural and home garden use of sewage sludge than is permitted by the U.S. EPA 503 rule.

Standardization of a technique for BrIDA cholescintigraphy.

Calculation of ejection fraction by cholescintigraphy is a well-established technique to quantify motor function of the gallbladder. All previous cholescintigraphic studies of gallbladder emptying have exclusively used the anterior projection for gallbladder visualization. This may lead to inaccuracies because attenuation of gamma rays may vary depending on the state of gallbladder filling and body habitus. The use of geometric means may prevent these errors. In this study, 7 health volunteers were examined after the administration of 150 MBq 99Tcm-bromotrimethyl-IDA (99Tcm-BrIDA) without any stimulus and 9 volunteers ingested a maxied, fatty meal 35-40 min after BrIDA injection. Imaging was done using a dualheaded camera in a dynamic study of 140 one-minute frames. A plateau of activity was reached between 70 and 90 min post-injection of BrIDA and between 55-60% of maximal counts were achieved in the gallbladder region of interest (ROI) between 35 and 40 min. Significant emptying was seen in all nine subjects after the meal. No significant difference in ejection fraction was seen between the anterior projection and the geometric mean of the data from the anterier and posterior projections. The slope of the gallbladder filling curve from the anterior camera data was different from that obtained from the geometric mean, but the correlation coefficient was not significantly different. In conclusion, adequate counts are achieved in the gallbladder ROI 35-40 min after BrIDA injection, when the stimulus can be administered to study gallbladder emptying. There is no advantage to imaging in the anterior and posterior projections and data obtained from the anterior projection alone are adequate for gallbladder emptying studies.

Comments on the interpretation of data from log normal distributions and definition of geometric standard deviation

AbstractEstimates of relative potency (commonly obtained as antilogs of log relative potencies) and the distribution of many pharmaceutical measurements are positively skewed and appear to follow a log normal distribution. The geometric mean arises naturally as a measure of location in these circumstances. There is not, however, a measure of dispersion which corresponds to the geometric mean and conflicting definitions have been used for the term geometric standard deviation. Examples of data for which the use of geometric means may be appropriate are given, together with comments on inferences which may be drawn regarding the population from which the data arise. It is suggested that the term geometric standard deviation leads to confusion and inappropriate inferences, and that it should therefore be avoided.

A nonlinear Runge-Kutta formula for initial value problems

New non-linear Runge-Kutta methods for solving initial value problems are shown to be obtained by the strategic use of geometric mean (GM) rather than arithmetic mean averaging of the functional values in the standard integration formula.

Rate-weight analysis: A critique and a reformulation

We have suggested a reformulation of the technique of DNO for regional-national growth rate differential by separate rate (growth) and weight effects as it is shown that their technique appears to be biased and inconsistent. From an analytical point of view the main points requiring modification are primarily two: (a) the use of geometric mean of initial and final year weights by sectors violates aggregation consistency, and (b) normalization rules are not obeyed by DNO for regional-national comparisons since uniform weights are not applied across both the region and the nation. An empirical test is performed by a reformulation (which eliminates the above two errors) with the employment data by 11 sectors for two regions, Atlantic Provinces and Ontario, vis-a-vis Canada (the nation) for two time units, 1961 and 1971. Separate rate and weight effects are then identified for Atlantic-Canada and Ontario-Canada.