Simple Mathematical Method Research Articles

Calculation of Equivalent Square Fields in Radiation Therapy by an Improved Vadash Correction Factor for Collimator Exchange Effects

Background and objectives:To determine the head scatter factor, a formalism presented by Vadash and Bjärngard has been employed to assess collimator exchange effects. The aim of this study was to determine the best Vadash correction factor (A) by introducing a new method based on the output factor measured in air for different square and rectangular fields.Materials and Methods:A new simple mathematical method based on selection of the best dosimetric data was proposed to obtain the A value for Vadash to predict the equivalent square field size. Measurements were performed with a Farmer chamber 0.6 cc in SSD 100 cm and build up cap of Plexiglas, 1.5 and 3.5 cm equal to water, for 6 and 18 MV, respectively, with a Perimus Plus linear accelerator. The output factor in air (OFair) was measured for square and rectangular fields. MATLAB software (version R2014a) was employed for calculations and curve fitting.Results:A power model with a constant value was applied to the output factor in air as a function of square field size. The output factor in air ranged from 0.983 to 1.038 at 6MV and from 0.731 to 1.05 at 18 MV with the Y collimator having a greater effect. Obtained values for A were 1.42 and 1.55 with respect to σmin=0.98 and 2.3 for 6 and 18 MV energies, respectively.Conclusion:The proposed A values minimize the collimator exchange effect in calculating equivalent squares, which plays an important role in patient dose calculation and treatment planning.

Studying associative learning without solving learning equations

I introduce a simple mathematical method to calculate the associative strengths of stimuli in many models of associative learning, without solving the models’ learning equations and without simulating the learning process. The method applies to many models, including the Rescorla and Wagner (1972) model, the replaced elements model of Brandon et al. (2000), and Pearce’s (1987) configural model. I illustrate the method by calculating the predictions of these three models in summation and blocking experiments, allowing for a degree of similarity between the training stimuli as well as for the effects of contextual stimuli. The method clarifies the models’ predictions and suggests new empirical tests.

A Simple Mathematical Method for Optimal Preliminary Design of Tall Buildings with Peak Lateral Deflection Constraint

The classic methods of structural design based on trial-and-error are generally highly iterative and computationally intensive, especially in a tall building consisting of thousands of structural members. This paper presents a closed-form solution for a minimum weight design problem which may be used at the early-stage design of a high-rise and slender structure. Before considering the weight-based problem, an intervening optimization formulation is introduced in advance which for a fixed amount of material presents an optimal pattern for its distribution. The obtained pattern is then used in the main problem, in which the total weight is considered as the objective function while satisfying the peak lateral deflection constraint and some practical requirements. The flexural stiffness in its general form is selected here as the design variable, so the proposed method is not restricted to a particular system, and any structural system may be addressed. A braced-tube 61-storey building example is presented to be designed based on the proposed technique to illustrate the effectiveness and practicality of the method.

A simple mathematical method to estimate ammonia emission from in-house windrowing of poultry litter

ABSTRACTIn-house windrowing between flocks is an emerging sanitary management practice to partially disinfect the built-up litter in broiler houses. However, this practice may also increase ammonia (NH3) emission from the litter due to the increase in litter temperature. The objectives of this study were to develop mathematical models to estimate NH3 emission rates from broiler houses practicing in-house windrowing between flocks. Equations to estimate mass-transfer areas form different shapes windrowed litter (triangular, rectangular, and semi-cylindrical prisms) were developed. Using these equations, the heights of windrows yielding the smallest mass-transfer area were estimated. Smaller mass-transfer area is preferred as it reduces both emission rates and heat loss. The heights yielding the minimum mass-transfer area were 0.8 and 0.5 m for triangular and rectangular windrows, respectively. Only one height (0.6 m) was theoretically possible for semi-cylindrical windrows because the base and the height were not independent. Mass-transfer areas were integrated with published process-based mathematical models to estimate the total house NH3 emission rates during in-house windrowing of poultry litter. The NH3 emission rate change calculated from the integrated model compared well with the observed values except for the very high NH3 initial emission rate from mechanically disturbing the litter to form the windrows. This approach can be used to conveniently estimate broiler house NH3 emission rates during in-house windrowing between flocks by simply measuring litter temperatures.

Control of Solar Power Plants Connected Grid with Simple Calculation Method on Residential Homes

One of the most compatible renewable energy in all regions to apply is solar energy. Solar power plants can be built connected to existing or stand-alone power grids. In assisting the residential electricity in which there is a power grid, then a small scale solar energy power plants is very appropriate. However, the general constraint of solar energy power plants is still low in terms of efficiency. Therefore, this study will explain how to control the power of solar power plants more optimally, which is expected to reactive power to zero to raise efficiency. This is a continuation of previous research using Newton Rapshon control method. In this study we introduce a simple method by using ordinary mathematical calculations of solar-related equations. In this model, 10 PV modules type of ND T060M1 with a 60 Wp capacity are used. The calculations performed using MATLAB Simulink provide excellent value. For PCC voltage values obtained a stable quantity of approximately 220 V. At a maximum irradiation condition of 1000 W / m2, the reactive power value of Q solar generating system maximum 20.48 Var and maximum active power of 417.5 W. In the condition of lower irradiation, value of reactive power Q almost close to zero 0.77Var. This simple mathematical method can provide excellent quality control power values.

Rock Stress Around Noncircular Tunnel: a New Simple Mathematical Method

AbstractA new simple mathematical method has been proposed to predict rock stress around a noncircular tunnel and the method is calibrated and validated with a numerical model. It can be found that the tunnel shapes and polar angles affect the applicable zone of the theoretical model significantly and the applicable zone of a rectangular tunnel was obtained using this method. The method can be used to predict the values of the concentrated stress, and to analyze the change rate of rock stress and back to calculate the mechanical boundary condition in the applicable zone. The results of the stress change rate indicate that the horizontal stress is negatively related to the vertical boundary load and positively related to the horizontal boundary load. The vertical stress is negatively related to the horizontal boundary load and positively related to the vertical boundary load. These findings can be used to explain the evolution of the vertical increment in stress obtained with field-based borehole stress monitoring.

Prediction of the backwater level due to bridge constriction in waterways

AbstractThis article proposes a mathematical method to calculate the backwater level at bridge constrictions for compound channels. Such a model is necessary, as many of the previously developed simple formulas yield inaccurate results. Parametric studies on a single‐opening semicircular arch, single‐opening semielliptical arch and single‐opening straight deck bridge were also conducted using the commercial software HEC‐RAS to investigate the influences of different factors on backwater level. The results indicated that the Froude number, opening ratio, discharge and roughness coefficient impact the backwater level considerably. Hence, a simple mathematical method was developed based on these parameters and validated using experimental data in a symmetrical two‐stage channel with different roughness coefficients for different types of bridge models. The results obtained using the proposed formula corresponded well with the experimental data and the results obtained using the energy method, which is accurate and the most commonly used method to calculate backwater level.

A mathematical method and artificial neural network modeling to simulate osmosis membrane’s performance

Lack of fresh water has been a major obstacle to development and flourishing in human history. Desalination provides a new vision toward fresh water production in the upcoming future. The study has proposed a simple mathematical equation and ANN models to simulate eight types of sea water RO membranes. The Artificial Neural Network (ANN) models have been developed to simulate TDS corresponding to the temperature (T, °C), flow rate (gpm) and recovery percentage. The feed data was generated by ROSA software. The model developed using a simple rational mathematical method. ANN models were trained using feed-forward back propagation algorithm with two hidden layers and various numbers of neurons in each layer. The model verification analysis proved both mathematical and ANN models to be highly accurate, reliable and practical for analyzing, designing, operating and optimizing of RO systems. The correlation coefficients (R) of 0.96 and 0.97, respectively, confirmed that the equation and ANN models resulted in this study are in good agreement with the measured data.

Real-time prediction of cell division timing in developing zebrafish embryo.

Combination of live-imaging and live-manipulation of developing embryos in vivo provides a useful tool to study developmental processes. Identification and selection of target cells for an in vivo live-manipulation are generally performed by experience- and knowledge-based decision-making of the observer. Computer-assisted live-prediction method would be an additional approach to facilitate the identification and selection of the appropriate target cells. Herein we report such a method using developing zebrafish embryos. We choose V2 neural progenitor cells in developing zebrafish embryo as their successive shape changes can be visualized in real-time in vivo. We developed a relatively simple mathematical method of describing cellular geometry of V2 cells to predict cell division-timing based on their successively changing shapes in vivo. Using quantitatively measured 4D live-imaging data, features of V2 cell-shape at each time point prior to division were extracted and a statistical model capturing the successive changes of the V2 cell-shape was developed. By applying sequential Bayesian inference method to the model, we successfully predicted division-timing of randomly selected individual V2 cells while the cell behavior was being live-imaged. This system could assist pre-selecting target cells desirable for real-time manipulation–thus, presenting a new opportunity for in vivo experimental systems.

Simple Mathematical Operation-Based Calibration Method for Giant Magnetoresistive Current Sensor Applying B-Spline Modeling

Nowadays, giant magnetoresistance (GMR)-based sensor technology is widely used in numerous industrial applications. However, an inexpensive and efficient method for compensating the nonlinear output of GMR spin valve sensors is still under the industrial prospect, which limits the application of the GMR sensors. By applying the B-Spline modeling, excellent results can be achieved; nevertheless, the sensor has to be calibrated in the whole temperature range. In this paper, a simple mathematical operation-based calibration method for GMR spin valve-type current sensors applying B-Spline modeling is introduced. Instead to measure dozens of temperatures and per temperature several hundreds of points, this method allows a calibration for each further sensor applying B-Spline modeling with only two temperatures and per temperature only 15 points. In comparison with the original calibration measurement over temperature for B-Spline modeling, the new calibration with minimized efforts shows outstanding measurement accuracy of ±0.6% at a nominal current of 50 A in a temperature range from −20 °C to 80 °C.

Active multi-scale modeling and gas permeability study of porous metal fiber sintered felt for proton exchange membrane fuel cells

Porous metal fiber sintered felt (PMFSF) is a promising critical component in proton exchange membrane fuel cells, having the ability of simultaneously acting as the flow field plate, gas diffusion layers and also the catalyst layers support, and owing the property of multi-scale surface morphology. A simple multi-scale mathematical method was proposed to actively construct three dimensional models of PMFSF’s microstructure by synthesizing the implicit periodic surface (PS) model and the Weierstrass-Mandelbrot (W-M) fractal geometry. In this method, the PS model described the macro overall fiber shape, and the W-M fractal geometry modeled the micro fractal roughness topography attached. Based on the method, multi-scale fractal PMFSF models were reconstructed according to morphology parameters of physical PMFSFs, and were discretized in ANSYS/ICEM to generate refined mesh for computational fluid dynamics analysis. To verify the validity of the proposed modeling approach, PMFSFs with different porosity and fiber orientation are generated, and then the effects of the fractal surface topography and the fractal parameters such as fractal dimension and height scaling parameter on the gas permeability of PMFSF were investigated. The numeric simulation results show that the influence of the fractal topography on the in-plane and through-plane permeability of PMFSF cannot be ignored, and the permeability of fractal PMFSF models agrees with experimental measurements better. Especially, the results imply that the fractal morphology may have the potential to adjust the anisotropic properties of PMFSFs’ permeability. It is further found that the larger the fractal dimension is and the lower the height scaling parameter is, the better the permeability of PMFSF will be. The synthesis approach and numerical simulation method may facilitate the development of active functional design mode to predict and optimize key characteristics of PMFSF ahead of manufacture.

Phase shifting in the spatial frequency domain

We present a simple mathematical method for phase shifting that overcomes some phase shift errors and limitations of commonly used methods. The method is used to generate a sequence of phase-shifted interferograms from a single interferogram. The generated interferograms are employed to reconstruct the wavefront aberrations, as an application. The approach yields results with only very small deviations compared to both simulated wavefront aberrations, including the first 25 Zernike polynomials (0.05%) and those measured with a Shack-Hartmann sensor (0.5%).

A Mathematical Method to Calculate Tumor Contact Surface Area: An Effective Parameter to Predict Renal Function after Partial Nephrectomy

We proposed a mathematical formula to calculate contact surface area between a tumor and renal parenchyma. We examined the applicability of using contact surface area to predict renal function after partial nephrectomy. We performed this retrospective study in patients who underwent partial nephrectomy between January 2012 and December 2014. Based on abdominopelvic computerized tomography or magnetic resonance imaging, we calculated the contact surface area using the formula (2*π*radius*depth) developed by integral calculus. We then evaluated the correlation between contact surface area and perioperative parameters, and compared contact surface area and R.E.N.A.L. (Radius/Exophytic/endophytic/Nearness to collecting system/Anterior/Location) score in predicting a reduction in renal function. Overall 35, 26 and 45 patients underwent partial nephrectomy with open, laparoscopic and robotic approaches, respectively. Mean ± SD contact surface area was 30.7±26.1 cm(2) and median (IQR) R.E.N.A.L. score was 7 (2.25). Spearman correlation analysis showed that contact surface area was significantly associated with estimated blood loss (p=0.04), operative time (p=0.04) and percent change in estimated glomerular filtration rate (p <0.001). On multivariate analysis contact surface area and R.E.N.A.L. score independently affected percent change in estimated glomerular filtration rate (p <0.001 and p=0.03, respectively). On ROC curve analysis contact surface area was a better independent predictor of a greater than 10% change in estimated glomerular filtration rate compared to R.E.N.A.L. score (AUC 0.86 vs 0.69). Using this simple mathematical method, contact surface area was associated with surgical outcomes. Compared to R.E.N.A.L. score, contact surface area was a better predictor of functional change after partial nephrectomy.

A Mathematical Method for Calculating the Anterior and Superior Axial Coverage Angle of Acetabulum to Femoral 5 Head based on 3D Coordinate System

Background: Direct imaging measurement of the anterior and superior axial coverage angle of acetabulum to femoral head is difficult. This study aimed to build the mathematical formulas to calculate them using the available parameters, e.g., acetabular abduction angle. Results: A 3D coordinate system was established by setting the center of the hip joint rotation basing as coordinate center O. In the drawn diagrams of anterior and superior axial coverage angle of acetabulum to femoral head, femoral neck anteversion was set as a, acetabular abduction angle as b, acetabular anteversion as A, and femoral neck-shaft angle as J. In the absence of the anteversion of acetabular and femoral neck, the superior axis coverage angle of acetabulum to the femoral head equals to b + 180-J. The anterior axial coverage angle of acetabulum to femoral head equals to 90 degrees. In the presence of the acetabular and femoral neck anteversion, the superior axis coverage angle of acetabulum to the femoral head equals to arctg {tgb / [cosacos (A + a)]} + 180-J; the anterior axis coverage angle of acetabulum to the femoral head equals to 90- arctg [tg (A + a) / cos {arctg [tgb / (cosa + a)] - (J-90)}]. The close relationship between the acetabulum and femoral head spatial orientation angle was observed in the absence of acetabular and femoral neck anteversion. The superior axis coverage angle of acetabulum to the femoral head was positively correlated with acetabular abduction angle and negatively with femoral neck-shaft angle. While the anterior axial coverage angle of acetabulum to femoral head was not dependent on acetabular abduction angle and femoral neck-shaft angle. In the presence of acetabular and femoral neck anteversion, both anterior and superior axial coverage angle of acetabulum to femoral head were relevant to acetabular abduction angle and femoral neck-shaft angle. The superior axial coverage angle of acetabulum to femoral head positively correlated with acetabular abduction angle, negatively with femoral neck-shaft angle, and positively with acetabular and femoral neck anteversion. The anterior axial coverage angle of acetabulum to femoral head was negatively correlated with acetabular abduction angle, positively with femoral neck-shaft angle and negatively with acetabular and femoral neck anteversion. Conclusions: The formulas were built based on a 3D coordinate system established in the current study. The formula reflects the inner relationship of spatial orientation angle of acetabulum to femoral head, which will offer a simple, novel and efficient mathematical method to calculate the anterior and superior axial coverage angle.

Planar hydrogen-like atom in inhomogeneous magnetic fields: Exactly or quasi-exactly solvable models

We use a simple mathematical method to solve the problem of a two-dimensional hydrogen-like atom in the inhomogeneous magnetic fields B = (k/r)z and B = (k/r3)z. We construct a Hamiltonian that takes the same form as the Hamiltonian of a hydrogen-like atom in the homogeneous magnetic fields and obtain the energy spectrum by comparing the Hamiltonians. The results show that the whole spectrum of the atom in the magnetic field B = (k/r)z can be obtained, and the problem is exactly solvable in this case. We find analytic solutions of the Schrodinger equation for the atom in the magnetic field B = (k/r3)z for particular values of the magnetic strength k and thus present a quasi-exactly solvable model.

Analysis and optimization of cross-immunity epidemic model on complex networks

There are various infectious diseases in real world, and these diseases often spread on a network of population and compete for the limited hosts. Cross-immunity is an important disease competing pattern, which has attracted the attention of many researchers. In this paper, we discovered an important conclusion for two cross-immunity epidemics on a network. When the infectious ability of the second epidemic takes a fixed value, the infectious ability of the first epidemic has an optimal value which minimizes the sum of the infection sizes of the two epidemics. We also proposed a simple mathematical analysis method for the infection size of the second epidemic using the cavity method. The proposed method and conclusion are verified by simulation results. Minor inaccuracies of the existing mathematical methods for the infection size of the second epidemic are also found and discussed in experiments, which have not been noticed in existing research.

The ratio of acetate-to-glucose oxidation in astrocytes from a single 13C NMR spectrum of cerebral cortex.

The (13) C-labeling patterns in glutamate and glutamine from brain tissue are quite different after infusion of a mixture of (13) C-enriched glucose and acetate. Two processes contribute to this observation, oxidation of acetate by astrocytes but not neurons, and preferential incorporation of α-ketoglutarate into glutamate in neurons, and incorporation of α-ketoglutarate into glutamine in astrocytes. The acetate:glucose ratio, introduced previously for analysis of a single (13) C NMR spectrum, provides a useful index of acetate and glucose oxidation in the brain tissue. However, quantitation of relative substrate oxidation at the cell compartment level has not been reported. A simple mathematical method is presented to quantify the ratio of acetate-to-glucose oxidation in astrocytes, based on the standard assumption that neurons do not oxidize acetate. Mice were infused with [1,2-(13) C]acetate and [1,6-(13) C]glucose, and proton decoupled (13) C NMR spectra of cortex extracts were acquired. A fit of those spectra to the model indicated that (13) C-labeled acetate and glucose contributed approximately equally to acetyl-CoA (0.96) in astrocytes. As this method relies on a single (13) C NMR spectrum, it can be readily applied to multiple physiologic and pathologic conditions. Differences in (13) C labeling of brain glutamate and glutamine have been attributed to metabolic compartmentation. The acetate:glucose ratio, introduced for description of a (13) C NMR (nuclear magnetic resonance) spectrum, is an index of glucose and acetate oxidation in brain tissue. A simple mathematical method is presented to quantify the ratio of acetate-to-glucose oxidation in astrocytes from a single NMR spectrum. As kinetic analysis is not required, the method is readily applicable to analysis of tissue extracts. α-KG=alpha-ketoglutarate; CAC=citric acid cycle; GLN=glutamine; GLU = glutamate.

Channel Capacity Delay Tradeoff for Two-Way Multiple-Hop MIMO Relay Systems with MAC–PHY Cross Layer

In multiple-hop MIMO relay systems, an end-to-end channel capacity is restricted by the bottleneck relay. Therefore, in order to obtain the high end-to-end channel capacity, we propose a simple mathematical method to optimize both distances and transmit powers simultaneously. Additionally, the end-to-end channel capacity of optimization based on an one-way and a two-ways transmissions is analyzed. The specific TDMA is proposed to control the transmission of all transmitters on MAC layer and then the distance and the transmit power are optimized based on MAC---PHY cross layer by the proposal particle filter method to obtain the higher end-to-end channel capacity. The calculation result indicates that there is the optimal number of relays that has the maximal end-to-end channel capacity and the trade-off between the end-to-end channel capacity and the delay time.

Bayesian Neural Network of Rolling Force Prediction for Hot-Strip Mill

For obtaining relative accurate rolling-mill model is difficulty by the simple mathematical method, due to the complexity of the actual production scene and the non-linear relationship between variables, this paper firstly proposes an improved Bayesian regularization neural network model according to these measured data of 1580 production line. In this model, the paper constructs the improved Bayesian neural networks by the introduction of bound terms that represents the network complexity in the objective function. At last, the simulation result proves the effectiveness and validity of the model and the prediction accuracy of the model algorithm is superior to the traditional model. DOI : http://dx.doi.org/10.11591/telkomnika.v12i5.5097

Binomial distribution for quantification of protein subunits in biological Nanoassemblies and functional nanomachines

Living systems produce ordered structures and nanomachines that inspire the development of biomimetic nanodevices such as chips, MEMS, actuators, sensors, sorters, and apparatuses for single-pore DNA sequencing, disease diagnosis, drug or therapeutic RNA delivery. Determination of the copy numbers of subunits that build these machines is challenging due to small size. Here we report a simple mathematical method to determine the stoichiometry, using phi29 DNA-packaging nanomotor as a model to elucidate the application of a formula ∑M=0ZZMpZ−MqM, where p and q are the percentage of wild-type and inactive mutant in the empirical assay; M is the copy numbers of mutant and Z is the stoichiometry in question. Variable ratios of mutants and wild-type were mixed to inhibit motor function. Empirical data were plotted over the theoretical curves to determine the stoichiometry and the value of K, which is the number of mutant needed in each machine to block the function, all based on the condition that wild-type and mutant are equal in binding affinity. Both Z and K from 1-12 were investigated. The data precisely confirmed that phi29 motor contains six copies (Z) of the motor ATPase gp16, and K=1. From the Clinical EditorTo determine copy numbers of subunits that form nanomachines in living organisms is a daunting task due to the complexities and the inherently small sizes associated with such systems. In this paper, a simple mathematical method is described how to determine the stoichiometry of copies in biomimetic nanodevices, using phi29 DNA-packaging nanomotor as a model.