Animal behavior analysis is a crucial tasks for the industrial farming. In an indoor farm setting, extracting Key joints of animal is essential for tracking the animal for longer period of time. In this paper, we proposed a deep network that exploit transfer learning to trained the network for the pig skeleton extraction in an end to end fashion. The backbone of the architecture is based on hourglass stacked dense-net. In order to train the network, key frames are selected from the test data using K-mean sampler. In total, 9 Keypoints are annotated that gives a brief detailed behavior analysis in the farm setting. Extensive experiments are conducted and the quantitative results show that the network has the potential of increasing the tracking performance by a substantial margin.

The aim of this study is (a) to develop, test, and employ a combined method of unsupervised machine learning to objectively assess the condition of sports facility in primary schools (PSSFC) and (b) examine the examine the geographical and typological association with PSSFC. Based on the Sixth National Sports Facility Census (NSFC), six PSSFC indicators (indoor and outdoor facility included) were selected as the measurements and decomposed by using the t-stochastic neighbor embedding (t-SNE). Thereafter, the Fuzzy C-mean (FCM) algorithm was used to cluster the same type of PSSFC with selecting the optimum numbers of evaluation level. Overall 845 primary schools in Shanghai, China were recruited and tested by this combined approach of unsupervised machine learning. In addition, the two-way analysis of covariance was used to examine the location and types of school associated with PSSFC variables in each level. The combined method was found to have acceptable reliability and good interpretability, differentiating PSSFC into five gradient levels. The characteristics of PSSFC differ by the location and school type of individual school. Our findings are conducive to the regionalized and personalized intervention and promotion on the children’s physical activity (PA) upon the practical situation of particular schools.

In the framework of 2D circular membrane Micro-Electric-Mechanical-Systems (MEMS), a new non-linear second-order differential model with singularity in the steady-state case is presented in this paper. In particular, starting from the fact that the electric field magnitude is locally proportional to the curvature of the membrane, the problem is formalized in terms of the mean curvature. Then, a result of existence of at least one solution is achieved. Finally, two different approaches prove that the uniqueness of the solutions is not ensured.

This paper is a continuation of previous papers on unification theories publishes in AXIOMS. We present results about the (modified) Yang-Baxter equation, Euler’s formula, dual numbers, coalgebra structures, non-associative structures, differential geometry, and (mean) inequalities. We will also attempt to relate our discussion to some Brain Studies and Machine Learnig.

Spatially uniform optical excitations can induce Floquet topological band structures within insulators which can develop similar or equal characteristics as are known from three-dimensional topological insulators. We derive in this article theoretically the development of Floquet topological quantum states for electromagnetically driven semiconductor bulk matter and we present results for the lifetime of these states and their occupation in the non-equilibrium. The direct physical impact of the mathematical precision of the Floquet-Keldysh theory is evident when we solve the driven system of a generalized Hubbard model with our framework of dynamical mean field theory (DMFT) in the non-equilibrium for a case of ZnO. The physical consequences of the topological non-equilibrium effects in our results for correlated systems are explained with their impact on optoelectronic applications.

This study demonstrates a global, non-parametric, non-iterative optimization of time-mean value of a kind of index vibrated by time-varying forcing. It is based on the fact that the (steady) forced vibration of non-autonomous ordinary differential equation systems is well approximated by an analytical solution when the amplitude of forcing is sufficiently small and its base state without forcing is linearly stable and steady. It is applied to optimize a time-averaged heat-transfer rate on a two-dimensional thermal convection field in a square cavity with horizontal temperature difference, and the globally optimal way of vibrational forcing, i.e. the globally optimal, spatial distribution of vibrational heat and vorticity sources, is first obtained. The maximized vibrational thermal convection corresponds well to the state of internal gravity wave resonance. In contrast, the minimized thermal convection is weak, keeping the boundary layers on both sidewalls thick.

The contribution of extra-tropical synoptic cyclones to the formation of summer-mean atmospheric circulation patterns in the Arctic is investigated by clustering the dominant Arctic circulation patterns by the self-organizing maps (SOMs) using the daily mean sea level pressure (MSLP) in the Arctic domain (≥ 60°N). Three SOM patterns are identified: one with prevalent low pressure anomalies in the Arctic Circle (SOM1) and two opposite dipoles with primary high pressure anomalies covering the Arctic Ocean (SOM2 and SOM3). The time series of summertime occurrence frequencies demonstrate the largest inter-annual variation in the SOM1, the slight decreasing trend in the SOM2, and the abrupt upswing after 2007 in the SOM3. The relevant analyses with produced cyclone track data confirm that the vital contribution. The Arctic cyclone activity is enhanced in the SOM1 because the meridional temperature gradient increases over the land–Arctic Ocean boundaries co-located with major extra-tropical cyclone pathways. The composite daily synoptic evolutions for each SOM reveal that the persistence of all the three SOMs is less than 5 days on average. These evolutionary short-term weather patterns have substantial variability at inter-annual and longer timescales. Therefore, the synoptic-scale activity is central to forming the seasonal-mean climate of the Arctic.

This paper proposed an explicit and simple representation of velocity fluctuation and the Reynolds stress tensor in terms of the mean velocity field. The proposed turbulence equations are closed. The proposed formulations reveal that the mean vorticity is the key source of producing turbulence. It is found that there are no velocity fluctuation and turbulence if there were no vorticity. As a natural consequence, the laminar- turbulence transition condition was obtained in a rational way.

The first detection of gypsum (CaSO4.2H2O) by the Mars Science Laboratory (MSL) rover Curiosity in the Gale Crater, Mars created a profound impact on planetary science and exploration. The unique capability of plasma spectroscopy involving in situ elemental analysis in extraterrestrial environments, suggesting the presence of water in the red planet based on phase characterization and providing a clue to Martian paleoclimate. The key to gypsum as an ideal paleoclimate proxy lies in its textural variants, and in this study terrestrial gypsum samples from varied locations and textural types have been analyzed by Laser Induced Breakdown Spectroscopy (LIBS) technique. Petrographic, sub-microscopic and powder X-ray diffraction characterizations confirm the presence of gypsum (hydrated calcium sulphate; CaSO4.2H2O), bassanite (semi-hydrated calcium sulphate; CaSO4.1/2H2O) and anhydrite (anhydrous calcium sulphate; CaSO4) along with accessory phases (quartz and jarosite). The principal component analysis of LIBS spectra from texturally varied gypsums can be differentiated from one another because of the chemical variability in their elemental concentrations. The concentration of gypsum is determined from the partial least-square regressions model. Rapid characterization of gypsum samples with LIBS is expected to work well in extraterrestrial environments.

Recently, the medical image processing is extensively used in several areas. In earlier detection and treatment of these diseases is very helpful to find out the abnormality issues in that image. Here there are number of methods available for segmentation to detect the lung nodule of computer tomography (CT) image. The main result of this paper, the earlier detection of lung nodules using Pre-processing techniques of top-hat transform, median and adaptive bilateral filter was compared both filtering methods and proved the adaptive bilateral filter is suitable method for CT images. The proposed segmentation technique uses novel strip method and the image is split into number of strips 3, 4, 5 and 6. A marker- watershed method based on PSO and Fuzzy C-mean Clustering method was proposed method. Firstly, the input image was reduced noise reduction and smoothing and the filter image is using strips method and then the image is segmented by marker watershed method. Secondly, the enhanced PSO technique was used to locate the better accurate value of the clustering centers of Fuzzy C-mean Clustering. Final stage, with the accurate value of centers and the enhanced target function and the small region of the segmented object was clustered by Fuzzy C-mean. In segmentation algorithm presented in this paper gives 95% of accuracy rate to detect lung nodules when strip count is 5.

In this paper, we propose a new mathematical model based on the association between susceptible and recovered individual, where the association between susceptible and recovered individual is disturbed by white noise. This model is based on demographic changes and is used for long term behavior. We study the stability of equilibria of the deterministic model and prove the conditions for the extinction of diseases. Then, we investigate and obtain the critical condition of the stochastic epidemic model for the extinction and the permanence in mean of the disease with the white noise. To verify our results, we present some numerical simulations for real data related to disease.

In the article, by applied the concept of strongly convex function and one known identity, we establish several Ostrowski type inequalities involving conformable fractional integrals. As applications, some new error estimations for the midpoint formula are provided as well.

The aim of this work is to study generalizations of the notion of mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the Weak Law of Large Numbers in the same way that the ordinary mean relates to the Strong Law. We propose a further generalization, also based on an improper integral, called the doubly weak mean, applicable to heavy-tailed distributions such as the Cauchy distribution and the other symmetric stable distributions, We also consider generalizations arising from Abel-Feynman type mollifiers that damp the behavior at infinity and alternative formulations of the mean in terms of the cumulative distribution and the characteristic function.

For many years, the programmable positioners have been widely applied in structures of modern electro-pneumatic final control elements. The positioner consists of an electro-pneumatic transducer, embedded controller and measuring instrumentation. Electro-pneumatic transducers that are used in positioners are characterized by a relatively short mean time-to-failure. The practical and economical method of a reasonable prolongation of this time is proposed in this paper. It is principally based on assessment and minimizing the effort of the embedded controller. For this purpose, were introduced: the control value variability, mean-time and the cumulative controller's effort. The diminishing of controller effort has significant practical repercussions, because it reduces the intensity of mechanical wear of the final control element components. On the other hand, the reduction of the cumulative effort is important in the context of process economy due to limitation of the consumption of energy of compressed air supplying the final control element. Therefore, the minimization of introduced effort factors has an impact on increasing the functional safety and economics of the controlled process. As a result of the performed simulations, the recommendations regarding the selection of the structure and tuning of positioner controller were elaborated. The simulations were performed in the Matlab-Simulink environment with the use of the liquid level control system in which a phenomenological model of a final control element was deployed. It has been proven that under appropriate conditions, it is possible to extend significantly the lifetime of the final control element and simultaneously enhance the control quality factors.

Using a theorem of Ulam and Hyers, we will prove the Hyers-Ulam stability of two-dimensional Lagrange's mean value points $(\eta, \xi)$ which satisfy the equation, $f(u, v) - f(p, q) = (u-p) f_x(\eta, \xi) + (v-q) f_y(\eta, \xi)$, where $(p, q)$ and $(u, v)$ are distinct points in the plane. Moreover, we introduce an efficient algorithm for applying our main result in practical use.

In this paper, mean-field type games between two players with backward stochastic dynamics are defined and studied. They make up a class of non-zero-sum differential games where the players' state dynamics solve backward stochastic differential equations (BSDEs) that depend on the marginal distributions of player states. Players try to minimize their individual cost functionals, also depending on the marginal state distributions. Under some regularity conditions, we derive necessary and sufficient conditions for existence of Nash equilibria. Player behavior is illustrated by numerical examples, and is compared to a centrally planned solution where the social cost, the sum of player costs, is minimized. The inefficiency of a Nash equilibrium, compared to socially optimal behavior, is quantified by the so-called price of anarchy. Numerical simulations of the price of anarchy indicate how the improvement in social cost achievable by a central planner depends on problem parameters.

This paper presents the methods of estimating the mean square error of hydrological forecasts, allowing for assessment of their practical applicability. Depending upon the amount and composition of available hydrometeorological data, an appropriate method for forecast error estimation is chosen. A system of statistical tests for comparison of different forecasting methods for the same hydrologic characteristic with the same lead time is presented. These tests allow for choosing an optimal and most accurate forecasting method. Hydrological forecasting method efficiency estimation is based on comparing the forecast error with climatology or inertial (persistence) forecast error using presented tests.

The Pythagorean fuzzy set (PFS), which is characterized by a membership and a non-membership degree and the square sum of them is less or equal to one, can act as an effective tool to express decision makers’ fuzziness and uncertainty. Considering that the Heronian mean (HM) is a powerful aggregation operator which can take the interrelationship between any two arguments, we study the HM in Pythagorean fuzzy environment and propose new operators for aggregating interval-valued Pythagorean fuzzy information. First, we investigate the HM and geometric HM (GHM) under interval-valued intuitionistic fuzzy environment and develop a series of aggregation operators for interval-valued intuitionistic fuzzy numbers (IVIFNs) including interval-valued intuitionistic fuzzy Heronian mean (IVIFHM), interval-valued intuitionistic fuzzy geometric Heronian mean (IVIFGHM), interval-valued intuitionistic fuzzy weighted Heronian mean (IVIFWHM) and interval-valued intuitionistic fuzzy weighted geometric Heronian mean (IVIFWGHM). Second, some desirable and important properties of these aggregation operators are discussed. Third, based on these aggregation operators, a novel approach to multi-attribute decision making (MADM) is proposed. Finally, to demonstrate the validity of the approach, a numerical example is provided and discussed. Moreover, we discuss several real-world applications of these operators within policy-making contexts.

Depending on human body conditions and environmental conditions, it is sometimes difficult to conduct subject experiments. In such cases, it is effective to use a thermal manikin. There are few studies that investigate the effect of the non-uniform and asymmetric outdoor thermal environment on the mean skin temperature. The purpose of this study is to clarify the influence of the non-uniform and asymmetric thermal radiation of short-wavelength solar radiation in an outdoor environment on the calculation of the mean skin temperature. The skin temperature of the front of the coronal surface, which was facing the sun and where the body received direct short-wavelength solar radiation, and the skin temperature of the rear of the coronal surface, which was in the shadow and did not receive direct short-wavelength solar radiation were respectively measured. The feet, upper arm, forearm, hand and lower leg, which are susceptible to short-wavelength solar radiation in a standing posture, had a noticeable difference in skin temperature between sites in the sun and in shade. The mean skin temperature of sites facing the sun was significantly higher than the mean skin temperature of those in the shade.

Based on author's previous work [Sun, B. The Reynolds Navier-Stokes Turbulence Equations of Incompressible Flow Are Closed Rather Than Unclosed. Preprints 2018, 2018060461 (doi: 10.20944/preprints201806.0461.v1)], this paper proposed an explicit representation of velocity fluctuation and formulated the Reynolds stress tensor in terms of the mean velocity field. The proposed closed Reynolds Navier-Stokes turbulence formulations reveal that the mean vorticity is the key source of producing turbulence.

Since the electro-hydraulic servo shaking table exists many nonlinear elements, such as, dead zone, friction and blacklash, its acceleration response has higher harmonics which result in acceleration harmonic distortion, when the electro-hydraulic system is excited by sinusoidal signal. For suppressing the harmonic distortion and precisely identify harmonics, a combination of the adaptive linear neural network and least mean M-estimate (ADALINE-LMM), is proposed to identify the amplitude and phase of each harmonic component. Namely, the Hampel’s three-part M-estimator is applied to provide thresholds for detecting and suppressing the error signal. Harmonic generators are used by this harmonic identification scheme to create input vectors and the value of the identified acceleration signal is subtracted from the true value of the system acceleration response to construct the criterion function. The weight vector of the ADALINE is updated iteratively by the LMM algorithm, and the amplitude and phase of each harmonic, even the results of harmonic components, can be computed directly online. The simulation and experiment are performed to validate the performance of the proposed algorithm. According to the experiment result, the above method of harmonic identification possesses great real-time performance and it has not only good convergence performance but also high identification precision.

We consider rotational hypersurface in the four dimensional Euclidean space. We calculate the mean curvature and the Gaussian curvature, and some relations of the rotational hypersurface. Moreover, we define the third Laplace-Beltrami operator and apply it to the rotational hypersurface.

Hamacher operation which is generalization of the Algebraic and Einstein operation, can widely provide a large number of arithmetical operation with respect to uncertainty information, and Heronian mean can deal with correlations of the input arguments or different criteria and don’t make calculation redundancy, meanwhile, the normal intuitionistic fuzzy numbers (NIFNs) can depict distinctively normal distribution information in practical decision making. In this paper, a multi-criteria group decision-making (MCGDM) problem is researched under the NIFNs environment, and a new MCGDM approach is introduced on the basis of the Hamacher operation. Firstly, according to Hamacher t-conorm and t-norm, some operational laws of NIFNs are presented. Secondly, it is noticed that Heronian mean can’t only once take into account mutual relation between attribute values once, but also consider the correlation between input argument and itself. Therefore, we develop some operators and study their properties in order to aggregate normal intuitionistic fuzzy numbers information, these operators include Hamacher Heronian mean (NIFHHM), Hamacher weighted Heronian mean (NIFHWHM), Hamacher geometric Heronian mean (NIFHGHM) and Hamacher weighted geometric Heronian mean (NIFHWGHM). Furthermore, we apply the proposed operators to the MCGDM problem and present a new method. The main characteristics of this new method involve that: (1) it is suitable to make decision under the normal intuitionistic fuzzy numbers environment and more reliable and reasonable to aggregate the normal distribution information. (2) it utilizes Hamacher operation which can provide more reliable and flexible decision-making results and offer an effective and powerful mathematic tool for the MAGDM under uncertainty. (3) it uses the Heronian mean operator which can considers relationships between the input arguments or the attributes and don’t brings subsequently about redundancy. Lastly, an application is given for showing the feasibility and effectiveness of the presented method in this paper.

The propeller jet from a ship has a significant component directed upwards towards the free surface of the water, which can be used for ice management. This paper describes a comprehensive laboratory experiment where the influences of operational factors affecting a propeller wake velocity field were investigated. The experiment was done on a steady wake field to investigate the characteristics of the axial velocity of the fluid in the wake and the corresponding variability downstream of the propeller. The axial velocities and the variability recorded were time-averaged. Propeller rotational speed was found to be the most significant factor, followed by propeller inclination. The experimental results also provide some idea about the change of the patterns of the mean axial velocity distribution against the factors considered for the test throughout the effective wake field, as well as the relationships to predict the axial velocity for known factors.

In this research paper, a hybrid Artificial Neural Network (ANN)-Fuzzy Logic Control (FLC) tuned Flower Pollination Algorithm (FPA) as a Maximum Power Point Tracker (MPPT) is employed to emend root mean square error (RMSE) of photovoltaic (PV) modeling. Moreover, Gaussian membership functions have been considered for fuzzy controller design. This paper interprets Luo converter occupied brushless DC motor (BLDC) directed PV water pump application. Experimental responses certify the effectiveness of the suggested motor-pump system supporting diverse operating states. Luo converter is newly developed dc-dc converter has high power density, better voltage gain transfer and superior output waveform and able to track optimal power from PV modules. For BLDC speed controlling there is no extra circuitry and phase current sensors are enforced for this scheme. The recentness of this attempt is adaptive neuro-fuzzy inference system (ANFIS)-FPA operated BLDC directed PV pump with advanced Luo converter has not been formerly conferred.

The main objective of this paper is to introduce a further extension of extended (p, q)-beta function by considering two Mittag-Leffler function in the kernel. We investigate various properties of this newly defined beta function such as integral representations, summation formulas and Mellin transform. We define extended beta distribution and its mean, variance and moment generating function with the help of extension of beta function. Also, we establish an extension of extended (p, q)-hypergeometric and (p, q)-confluent hypergeometric functions by using the extension of beta function. Various properties of newly defined extended hypergeometric and confluent hypergeometric functions such as integral representations, Mellin transformations, differentiation formulas, transformation and summation formulas are investigated.

The concept of neutrosophic number is a significant mathematical tool to deal with real scientific problems because it can tackle indeterminate and incomplete information which exists generally in real problems. In this article, we use neutrosophic numbers (a + bI), where a and bI denote determinate component and indeterminate component respectively. We explore the situations in which the input information is needed to express in terms of neutrosophic numbers. We define score functions and accuracy functions for ranking neutrosophic numbers. We then define a cosine function to determine unknown criteria weights. We define neutrosophic number harmonic mean operators and proved their basic properties. Then, we develop two novel MCGDM strategies using the proposed aggregation operators. We solve a numerical example to demonstrate the feasibility and effectiveness of the proposed two strategies. Sensitivity analysis with variation of “I” on neutrosophic numbers is performed to demonstrate how the preference ranking order of alternatives is sensitive to the change of “I”. The efficiency of the developed strategies is ascertained by comparing the obtained results from the proposed strategies with the existing strategies in the literature.

Accurate and timely rainfall prediction enhances productivity and can aid proper planning in sectors such as agriculture, health, transport and water resources. This study is aimed at improving rainfall prediction using ensemble methods. It first assesses the performance of six convective schemes (Kain–Fritsch (KF); Betts–Miller–Janji´c (BMJ); Grell–Fretas (GF); Grell 3D ensemble (G3); New–Tiedke (NT) and Grell–Devenyi (GD)) using the root mean square error (RMSE) and mean error (ME) focusing on the March–May 2013 rainfall period over Uganda. 18 ensemble members are generated from the three best performing convective schemes (i.e. KF, GF & G3). The performance of three ensemble methods (i.e. ensemble mean (EM); ensemble mean analogue (EMA) and multi–member analogue ensemble (MAEM)) is also analyzed using the RMSE and ME. The EM presented a smaller RMSE compared to individual schemes (EM:10.02; KF:23.96; BMJ:26.04; GF:25.85; G3:24.07; NT:29.13 & GD:26.27) and a better bias (EM:-1.28; KF:-1.62; BMJ:-4.04; GF:-3.90; G3:-3.62; NT:-5.41 & GD:-4.07). The EMA and MAEM presented 13 out of 21 stations & 17 out of 21 stations respectively with smaller RMSE compared to EM thus demonstrating additional improvement in predictive performance. The MAEM is a new approach proposed and described in the study.

This paper presents the results of an experimental investigation about target detecting with passive sonar in Persian Gulf. Detecting propagated sounds in the water is one of the basic challenges of the researchers in sonar field. This challenge will be complex in shallow water (like Persian Gulf) and noise less vessels. Generally, in passive sonar the targets are detected by sonar equation (with constant threshold) which increase the detection error in shallow water. Purpose of this study is proposed a new method for detecting targets in passive sonars using adaptive threshold. In this method, target signal (sound) is processed in time and frequency domain. For classifying, Bayesian classification is used and prior distribution is estimated by Maximum Likelihood algorithm. Finally, target was detected by combining the detection points in both domains using LMS adaptive filter. Results of this paper has showed that proposed method has improved true detection rate about 27% compare other the best detection method.

The main focus when developing software is to improve the reliability and stability of a software system. When software systems are introduced, these systems are used in field environments that are the same as or close to those used in the development-testing environment; however, they may also be used in many different locations that may differ from the environment in which they were developed and tested. In this paper, we propose a new software reliability model that takes into account the uncertainty of operating environments. The explicit mean value function solution for the proposed model is presented. Examples are presented to illustrate the goodness-of-fit of the proposed model and several existing non-homogeneous Poisson process (NHPP) models and confidence intervals of all models based on two sets of failure data collected from software applications. The results show that the proposed model fits the data more closely than other existing NHPP models to a significant extent.

Kelly's Criterion is well known among gamblers and investors as a method for maximizing the returns one would expect to observe over long periods of betting or investing. These ideas are conspicuously absent from portfolio optimization problems in the financial and automation literature. This paper will show how Kelly's Criterion can be incorporated into standard portfolio optimization models. The model developed here combines risk and return into a single objective function by incorporating a risk parameter. This model is then solved for a portfolio of 10 stocks from a major stock exchange using a differential evolution algorithm. Monte Carlo calculations are used to verify the accuracy of the results obtained from differential evolution. The results show that evolutionary algorithms can be successfully applied to solve a portfolio optimization problem where returns are calculated by applying Kelly's Criterion to each of the assets in the portfolio.

An approach to the assessment of probabilistic inference is described which quantifies the performance on the probability scale. From both information and Bayesian theory, the central tendency of an inference is proven to be the geometric mean of the probabilities reported for the actual outcome and is referred to as the “Accuracy.” Upper and lower error bars on the accuracy are provided by the arithmetic mean and the -2/3 mean. The arithmetic is called the “Decisiveness” due to its similarity with the cost of a decision and the -2/3 mean is called the “Robustness”, due to its sensitivity to outlier errors. Visualization of inference performance is facilitated by plotting the reported model probabilities versus the histogram calculated source probabilities. The visualization of the calibration between model and source is summarized on both axes by the arithmetic, geometric, and -2/3 means. From information theory, the performance of the inference is related to the cross-entropy between the model and source distribution. Just as cross-entropy is the sum of the entropy and the divergence; the accuracy of a model can be decomposed into a component due to the source uncertainty and the divergence between the source and model. Translated to the probability domain these quantities are plotted as the average model probability versus the average source probability. The divergence probability is the average model probability divided by the average source probability. When an inference is over/under-confident, the arithmetic mean of the model increases/decreases, while the -2/3 mean decreases/increases, respectively.

In the paper, the authors survey integral representations (including the Lévy--Khintchine representations) and applications of some bivariate means (including the logarithmic mean, the identric mean, Stolarsky's mean, the harmonic mean, the (weighted) geometric means and their reciprocals, and the Toader--Qi mean) and the multivariate (weighted) geometric means and their reciprocals, derive integral representations of bivariate complex geometric mean and its reciprocal, and apply these newly-derived integral representations to establish integral representations of Heronian mean of power 2 and its reciprocal.

In the paper, by virtue of a Lévy–Khintchine representation and an alternative integral representation for the weighted geometric mean, the authors establish a Lévy–Khintchine representation and an alternative integral representation for the Toader–Qi mean. Moreover, the authors also collect an probabilistic interpretation and applications in engineering of the Toader–Qi mean.

Abstract: The aim of this study was to find out the incidence of anemia in pregnant women of Swat District; to analyze the iron variations and its dietary effects.Data were collected during the periods of January – September 2016. The study of samples comprised of 250 pregnant women in the different trimester. Blood sample from each woman was collected and full blood count (FBC) was conducted through Mindray BC-3000 plus hem analyzer for all pregnant individuals. Confirmed anemic cases were then examined for IDA with serum ferritin, serum iron, total iron binding capacity (TIBC) through Randox kit and serum transferrin saturation was estimated by formula (serum ferritin saturation =serum iron ×100/TIBC). The total number of participants in the first trimester were 50, among them 26 women were suffer from iron deficiency anemia (IDA) with 52% weightage of prevalence rate, (mean Hb concentration 9.602 ± 0.87 g/dl). The rates of IDA were 63.3%; ( mean Hb concentration 8.48 ± 1.24 g/dl) and 54%; ( mean Hb concentration 9.18 ± 1.28 g/dl), among 150 and 50 participants in the second and third trimester, respectively. A significant correlation was found between serum ferritin and Hb, serum ferritin against MCV and serum ferritin against MCH. The high prevalence of anemia was found 78.2% in the age group from 26-30 followed by 78.2% in the age group 36-40 years compared to those of other age groups in the second trimester. In this study the prevalence of IDA in third trimester is lower compared to first and second trimester.