On some mappings associated with geometric and arithmetic means

We are all familiar with the average, or arithmetic mean, of two numbers. Less frequently used is the notion of the geometric mean. In “Geometric Meaning in the Geometric Mean Means More Meaningful Mathematics” in the March 2001 issue of the Mathematics Teacher, Matt E. Fluster shows how the geometric mean, s = _ab, of two positive numbers, a and b, can be used in a first-year algebra course to tie together geometric, algebraic, and computational investigations. In this article, I add a bit of history and an example suitable for more advanced courses. The example uses the geometric mean to compute square roots with Newton's method and does not require calculus. The history surrounding these concepts provides an opportunity to point out the universal nature of mathematics as a human activity. The basic ideas in this article were alive in the minds of ancient peoples who lived in what are now India, Pakistan, Iraq, and Egypt.

Twelve healthy dogs were studied in this parallel group, blinded, randomised, and negative controlled efficacy study. On Day -1, the 12 dogs included were ranked within sex in descending order of individual pre-treatment (Day -5) fed mosquito counts and randomly allocated by blocks of two dogs to the untreated control group or the afoxolaner-treated group. NexGard® (Merial, now part of Boehringer Ingelheim Animal Health) was administered orally on Day 0 in accordance with the European label instructions. On Days 1, 7, 14, 21 and 28, all dogs were exposed for a duration of 1 hour to 50 ± 5 unfed Aedes aegypti females. After each exposure, mosquitoes were collected after 1 hour and assessed for viability during collection and at 24 ± 2 hours. The arithmetic (and geometric) mean values of live fed mosquito counts at 24 hours after the exposure periods for the negative control group ranged from 33.7 (32.3) to 49.8 (49.7), indicating that this was a vigorous mosquito strain. There was no significant difference between control and treated groups in the number of live and fed mosquitoes at each 1 hour post-exposure collection time. Based on arithmetic and geometric mean values at 24 hours after each exposure, significantly fewer live fed mosquitoes were recorded in the treated group, compared to the negative control group, throughout the study (p < 0.001). The afoxolaner insecticidal efficacy against A. aegypti varied from 98% (Day 2) to 75.3% (Day 29) based on arithmetic means, and 98.7% (Day 2) to 89.8% (Day 29) based on geometric means.

Maternal nutritional status during pregnancy and surma use determine cord lead levels in Karachi, Pakistan

Nadaraya-Watson kernel regression estimator (NWKRE) is a typical kernel regression estimator which is a kernel-based and non-parametric regression method to estimate the conditional expectation of a random variable and the non-linear mapping from input to output. For NWKRE, the selection of bandwidth, i.e., smoothing parameter h, plays a very important role in the fitting performance. In order to enhance the performance of NWKRE, an adaptive Nadaraya-Watson kernel regression estimator is proposed, ANWKRE for short. There are two main strategies to determine the adaptive or local bandwidth factor λ: geometric mean and arithmetic mean based determination methods, respectively. In this paper, we firstly investigate the mathematical properties of geometric mean and arithmetic mean in the framework of regression analysis. Then, some experimental comparisons are conducted to demonstrate our theoretical results. The experimental results find that the arithmetic mean based ANWKRE can obtain a smoother regression estimation for unknown function.

The difficulty of establishing a (noncommutative) matrix inequality involving the geometric mean was discussed in 1978 by K.V. Bhagwat and R. Subramanian [9] who pointed out that the problem of defining a geometric mean for non-commutative operators “makes it difficult to establish the validity or otherwise of the classical inequalities involving the geometric mean”. However, in a recent paper, Sagae and Tanabe [32] define a geometric mean and establish an AG-GM inequality for a finite number of positive definite matrices.

Methane (CH₄) emissions and oxidation were measured at the Air Hitam sanitary landfill in Malaysia and were modeled using the Intergovernmental Panel on Climate Change waste model to estimate the CH₄ generation rate constant, k. The emissions were measured at several locations using a fabricated static flux chamber. A combination of gas concentrations in soil profiles and surface CH₄ and carbon dioxide (CO₂) emissions at four monitoring locations were used to estimate the CH₄ oxidation capacity. The temporal variations in CH₄ and CO₂ emissions were also investigated in this study. Geospatial means using point kriging and inverse distance weight (IDW), as well as arithmetic and geometric means, were used to estimate total CH₄ emissions. The point kriging, IDW, and arithmetic means were almost identical and were two times higher than the geometric mean. The CH₄ emission geospatial means estimated using the kriging and IDW methods were 30.81 and 30.49 gm(−2) day(−1), respectively. The total CH₄ emissions from the studied area were 53.8 kg day(−1). The mean of the CH₄ oxidation capacity was 27.5 %. The estimated value of k is 0.138 year(−1). Special consideration must be given to the CH₄ oxidation in the wet tropical climate for enhancing CH₄ emission reduction.

In 1979, a study indicated that childhood cancers might be linked to extremely low frequency (ELF) magnetic field (Wertheimer, 1979). Leaning on numerous epidemiological studies, the International Agency for Research on Cancer classified in 2001 ELF magnetic field as possibly carcinogenic to human. These conclusions were based on a statistical association found in some epidemiological studies, unconfirmed by experimental results, between childhood leukaemia risk and a mean exposure over 24h higher than 0.4µT. The French population exposure to 50Hz magnetic field is not known. The exposure inside houses near power lines was assessed in the Cote-d’Or department (Clinard, 2004). But it was the buildings’ exposure and not the persons’ one. So the Direction Generale de la Sante (French Health Ministry) initiated a study on the French population exposure, named EXPERS. This communication gives preliminary results of the first measurement campaign in the Ile-de-France and Rhone-Alpes regions. A sample of 551 voluntary persons has been randomly taken from telephone lists. Each one has worn an EMDEX II (Enertech, USA) recording magnetic field (MF) each 3 s during 24h, has filled a timetable corresponding to the time of measurement and a questionnaire with informations on his socioprofessional data and on his house. We present here results on 458 persons, which can be split into 209 children (0-14 years) and 249 adults. The other have been withdrawn from this analysis because their measurements were not corresponding entirely to their exposures. The observed arithmetic and geometric means are respectively 0.05µT and 0.03µT for children, 0.08µT and 0.04µT for adults. Two aspects of the exposure have been particularly studied: - the search for the most discriminating MF descriptors from a hierarchical clustering classification followed by CART method (Classification And Regression Tree). This led to a distribution of the persons into two groups. The most discriminating chosen descriptors are the arithmetic mean, the geometric mean and the median. - the identification, by linear and logistic regression methods, of the factors that lead a person to be more exposed. These factors are: o for children: to live in a building o for adults: to live in a city with more than 2.000 habitants, to spend time in rail transport and in shopping centers. From the house addresses, the transmission and distribution system operators (RTE and ERDF) will indicate the presence of electric installations.

Microphone array techniques are an efficient tool to detect acoustic source positions. The delay and sum beamforming is the standard method. In the time domain, the generalized cross-correlation can be used to compute the noise source map. This technique is based on the arithmetic mean of the spatial likelihood functions. In this study, the classical arithmetic mean is replaced by the more standard generalized mean. The noise source maps provide by the arithmetic, geometric and harmonic means are compared in the case of numerical and experimental data obtained in a reverberant room. The geometric and harmonic means provide the best noise source maps with no side lobes and a better source level estimation.

In this paper, the characteristics of electric fields due to preliminary breakdown pulses (PBP) that precede the first return strokes (FRS) are presented. These pulses exist due to the electrical activity formed inside the clouds prior to the occurrence of the stepped leader. The electric field signatures were recorded in Bogota, Colombia with an altitude of 2550 m. For the measurement campaign, a broadband parallel-plate antenna system with a 500 ms full observation window and a pre-trigger of 250 ms were used. The measurements were carried out during two rainy seasons recording 180 waveforms. From recorded data, it was found that 37.5% of negative cloud-to-ground (CG) flashes exhibited PBP. With respect to the PBP trains a duration between 0.4 ms and 22.3 ms was observed. For the same temporal parameter, the arithmetic mean (AM) and the geometric mean (GM) were 4.7 ms and 3.1 ms, respectively. On the other hand, the GM of the time interval between the largest peak of the PBP train and the peak value of the FRS was 10.1 ms, while its AM was 27.9 ms. In addition, the GM of the ratio between the maximum value of the PBP train and its respective FRS was 0.51, with changes between 0.12 (minimum) and 2.1 (maximum). Finally, the number of individual pulses per train between 4 and 39.

Radon, the primary constituent of natural radiation, is the second leading environmental cause of lung cancer after smoking. To confirm a relationship between indoor radon exposure and lung cancer, estimating cumulative levels of exposure to indoor radon for an individual or population is necessary. This study sought to develop a model for estimate indoor radon concentrations in Korea. Especially, our model and method may have wider application to other residences, not to specific site, and can be used in situations where actual measurements for input variables are lacking. In order to develop a model, indoor radon concentrations were measured at 196 ground floor residences using passive alpha-track detectors between January and April 2016. The arithmetic mean (AM) and geometric mean (GM) means of indoor radon concentrations were 117.86±72.03 and 95.13±2.02 Bq/m3, respectively. Questionnaires were administered to assess the characteristics of each residence, the environment around the measuring equipment, and lifestyles of the residents. Also, national data on indoor radon concentrations at 7643 detached houses for 2011-2014 were reviewed to determine radon concentrations in the soil, and meteorological data on temperature and wind speed were utilized to approximate ventilation rates. The estimated ventilation rates and radon exhalation rates from the soil varied from 0.18 to 0.98/hr (AM, 0.59±0.17/hr) and 326.33 to 1392.77 Bq/m2/hr (AM, 777.45±257.39; GM, 735.67±1.40 Bq/m2/hr), respectively. With these results, the developed model was applied to estimate indoor radon concentrations for 157 residences (80% of all 196 residences), which were randomly sampled. The results were in better agreement for Gyeonggi and Seoul than for other regions of Korea. Overall, the actual and estimated radon concentrations were in better agreement, except for a few low-concentration residences.

Modeling the Risk of Salmonellosis from Consumption of Peanuts in the United States

This dissertation is concerned with the application of notions and methods from the field of information theory to the field of communication complexity. It consists of two main parts. In the first part of the dissertation, we prove lower bounds on the randomized two-party communication complexity of functions that arise from read-once boolean formulae. A read-once boolean formula is a formula in propositional logic with the property that every variable appears exactly once. Such a formula can be represented by a tree, where the leaves correspond to variables, and the internal nodes are labeled by binary connectives. Under certain assumptions, this representation is unique. Thus, one can define the depth of a formula as the depth of the tree that represents it. The complexity of the evaluation of general read-once formulae has attracted interest mainly in the decision tree model. In the communication complexity model many interesting results deal with specific read-once formulae, such as disjointness and tribes. In this dissertation we use information theory methods to prove lower bounds that hold for any read-once formula. Our lower bounds are of the form n(f )/c^d(f ) , where n(f ) is the number of variables and d(f ) is the depth of the formula, and they are optimal up to the constant in the base of the denominator. In the second part of the dissertation, we explore the applicability of the information-theoretic method in the number-on-the-forehead model. The work of Bar-Yossef, Jayram, Kumar &amp; Sivakumar [BYJKS04] revealed a beautiful connection between Hellinger distance and two-party randomized communication protocols. Inspired by their work and motivated by the open questions in the number-on-the-forehead model, we introduce the notion of Hellinger volume. We show that it lower bounds the information cost of multi-party protocols. We provide a small toolbox that allows one to manipulate several Hellinger volume terms and also to lower bound a Hellinger volume when the distributions involved satisfy certain conditions. In doing so, we prove a new upper bound on the difference between the arithmetic mean and the geometric mean in terms of relative entropy. Finally, we show how to apply the new tools to obtain a lower bound on the informational complexity of the ANDk function.

Michigan water quality standards for public bathing beaches require local health departments to collect and analyze a minimum of three water samples for Escherichia coli during each sampling event. The geometric mean number of E. coli colonies is then compared to the 300 colonies per 100 ml standard to determine compliance. This article compares the results of the currently mandated procedure to a composite sampling method, whereby the three samples are mixed in equal volumes and analyzed once. This effectively replaces the geometric mean of the individual sample results with an arithmetic mean. Although arithmetic means are more affected by outliers, this sensitivity to high concentrations is more health conservative than the geometric mean. During the 2007 sampling season, nine bathing beaches were monitored once each week. Three individual point samples and a composite sample were analyzed for each sampling event. No statistically significant differences in bacteria concentrations were found between composite sample analysis and the arithmetic mean of individual point sample analyses. No violations were detected in the 2007 sampling season, so using historical data, a retrospective analysis was performed on samples gathered at nine bathing beaches in Kalamazoo County, Michigan during the years 2001-2007. The arithmetic mean of the three samples taken at each site served as a surrogate composite sample. The benefits of compositing the three samples were investigated assuming a 2/3 reduction in analytical costs. In the traditional sampling method, three individual samples were obtained and analyzed once in every 3-week period during the summer season, whereas compositing was simulated by taking the arithmetic mean of each week's results. The results of this retrospective cost analysis indicates that ten to 14 violations would have been missed using the less frequent traditional sampling and analysis methodology. Composite sampling is a cost-saving alternative to traditional sampling techniques that can be more protective of public health, particularly when the savings are applied to increased numbers of samples in time or space.

Our recent study shows that Hyperion data and Hyperspectral Optimization Process Exemplar (HOPE) can effectively retrieve the shallow water depth in the South China Sea. In this study, the HJ-1A/HSI and EO-1/Hyperion remote sensing reflectance (R rs ) is simulated by the ocean radiative transfer software Hydrolight and used to derive shallow water depth with HOPE. Data merging is simulated in four different types of sea water inherent optical properties and four different bottom types. The results show Lee's weight mean theory performs obviously better than arithmetic mean, geometric mean in different conditions, which relates to bottom reflectance and seawater inherent optical properties. We further find that the difference between arithmetic mean, geometric mean and weighted mean decrease with the increase of water depth in the range of 0 to 20m and which depends on the bottom reflectance and seawater inherent optical properties. It will contribute to data merging of shallow water depth derived from hyperspectral data.

In this paper, statistical averaging method (arithmetic mean, geometric mean) and new statistical averaging method (new arithmetic mean, new geometric mean) have been proposed for extreme point multi-objective linear programming problem (EPMOLPP). Extreme point can be taken from graphical representation of linear programming problem (LPP). Graphical solution of LPP has been discussed in this research The objective of this method is for making single objective from multi-objective extreme point linear programming problem. Chandra Sen’s method is for making single objective from multi-objective linear programming problem (MOLPP). Here Chandra Sen’s method has also been used to solve EPMOLPP. An algorithm and program solution have been given for our proposed method to solve such type of problems. A numerical example is given and the result in Table 2 indicates that the proposed technique gives better results.