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A conventional oil and gas well does not have a natural production capacity, which necessitates a hydraulic fracturing operation. The effectiveness of the fracturing directly impacts the economic benefit of a single well. Among the various parameters, including fracture spacing, fracture width, and conductivity, fracture half-length is one of the main influencing factors on the productivity of horizontal wells. For conventional homogeneous reservoirs, research mainly focuses on fracture patterns with equal fracture lengths. However, in actual production processes, due to mutual interference and the superimposition of drainage areas between fractures, the production distribution of each fracture is non-uniform. Typical fracture distribution patterns mainly include uniform, staggered, dumbbell, and spindle. While many believe that the dumbbell-shaped fracture distribution pattern has the best effect, there has been no quantitative study on the length of each fracture under the dumbbell-shaped pattern. Based on this, this paper proposes a modeling approach for function-shaped fracture distribution that takes advantage of the high production of edge fractures and the low output of middle fractures in horizontal wells. The influence of this approach on production capacity is studied. Constant, linear, and parabolic functions are used to establish the relationship between fracture position and fracture half-length, optimizing the fracture distribution function to achieve the best production effect. This method can guide the horizontal well fracture distribution in the block to maximize productivity. The results show that the parabolic function-shaped model is better than the linear function-shaped model and the constant function-shaped model is the least effective. The research presented in this paper offers a new idea for optimizing on-site fracturing plans. It utilizes mathematical expressions to describe the parameters that affect productivity, which provides valuable guidance for designing multi-stage fractured horizontal wells in the field. In the future, this research will be extended by exploring the optimal fracture distribution function under different formation conditions.

У статті наведено одержані результати проведених польових експериментальних випробувань модернізованої коренезбиральної машини МКК-6, яку обладнано комбінованою системою для очистки кормових буряків від різного роду забруднень при їх збиранні. На основі експериментальних результатів виведено рівняння регресії, які характеризують зміну загальної забрудненості вороху коренеплодів різними домішками та побудовано графічні залежності загальної забрудненості вороху кормових буряків, пошкоджень коренеплодів, забрудненості коренеплодів рослинними рештками і кількості налиплого ґрунту на поверхні тіла коренеплодів залежно від зміни кутової швидкості обертання гвинта та діаметра очисних пружних елементів. При аналізі залежностей виявлено, що загальна забрудненість вороху коренеплодів залежно від кутової швидкості обертання гвинта змінюється за параболічною функцією, яка має яскраво виражений оптимум. Також зв’ясовано, що основним фактором, який впливає на кількість налиплого ґрунту на поверхні тіла коренеплодів є діаметр очисних пружних елементів. При його збільшенні кількість налиплого ґрунту на поверхні тіла коренеплодів значно зменшується при всіх значеннях кутової швидкості обертання гвинта.

The isospin-asymmetry nuclear chains with $N\ensuremath{-}Z$ from 2 to 60 are investigated to find the correlations of $\ensuremath{\alpha}$-decay properties and isospin asymmetry. Both the proton- and neutron-skin thicknesses of $\ensuremath{\alpha}$ emitters belonging to different $N\ensuremath{-}Z$ chains are tightly aligned along straight lines, the slope of which depends on the shell closure. For even-even $\ensuremath{\alpha}$ emitters, the neutron-skin thickness ${\mathrm{\ensuremath{\Delta}}}_{n}$ multiplied by $N\ensuremath{-}Z$ is a parabolic function of the isospin-asymmetry parameter $I=(N\ensuremath{-}Z)/A$. For the nuclei of incomplete neutron shells, the $\ensuremath{\alpha}$-decay half-life ${T}_{\ensuremath{\alpha}}$ exponentially increases with ${\mathrm{\ensuremath{\Delta}}}_{n}$ in the isospin chain. In the nuclei with valence neutrons and/or protons outside the magic core, ${T}_{\ensuremath{\alpha}}$ exponentially decreases with increasing ${\mathrm{\ensuremath{\Delta}}}_{n}$. The values of ${T}_{\ensuremath{\alpha}}$ and ${Q}_{\ensuremath{\alpha}}$ are almost exponential functions of isospin asymmetry.

This paper deals with GPU computing of special mathematical functions that are used in Fractional Calculus. The graphics processing unit (GPU) has grown to be an integral part of nowadays’s mainstream computing structures. The special mathematical functions are an integral part of Fractional Calculus. This paper deals with a novel parallel approach for computing special mathematical functions used in Fractional Calculus. NVIDIA’s GPU hardware is used to speed up the parallel algorithm. A comparison of the sequential code, vectorized code and GPU code is performed. We have successfully reduced the computation time of special mathematical functions using the parallel computing capabilities of GPU.

To optimize the laying length of the thin-walled micro-spraying belt, the field layout way of the micro-spraying belt was considered to determine the water head distribution coefficient β as 0.55 in lateral pipe. The allowable water head deviation of the micro-spraying belt was calculated. Combined with the energy conservation law, the models of the relative pressure function between the head end and the tail end of the micro-spraying belt under different slope-laying conditions were established, so were the mathematical relationships between the allowable head deviation of the micro-spraying belt and, the head end and the tail end relative pressure. Thus, the optimization of the laying length and slope was analyzed. It is found that the relationship between the relative pressure and the laying length is a decreasing function at the adverse slope while a parabolic function at the down slope. When the laying length of the micro-spraying belt is arranged to L0, the relative pressure P (L0) gets the maximum value. The paper put forward the laying length formula of the micro-spraying belt under different laying conditions—down slope, flat slope and adverse slope. The laying length of down slope of the micro-spraying belt with fold diameter of 43 and 64 mm was calculated under the condition of adverse slope at an incline of 0.01, 0.03 and 0.05 degrees and flat slope, respectively, aiming to provide a theoretical basis for engineering plan and design.

In this paper we introduce a generalization of the Voigt functions and discuss their properties and applications. Some interesting explicit series representations, integrals and identities and their link to Jacobi,Laguerre and Hermite polynomials are obtained. The resulting formulas allow a considerable unification of various special results which appear in the literature. © 2019. Acad. Colomb. Cienc. Ex. Fis. Nat. Resumen En este artículo se introduce una generalización de las funciones de Voigt y se discuten sus propiedades y aplicaciones. Se obtienen representaciones explícitas de series, integrales e identidades y sus conexiones con los polinomios de Jacobi, Laguerre y Hermite. Las fórmulas resultantes permiten la unificación de algunos resultados especiales que aparecen en la literatura. © 2019. Acad. Colomb. Cienc. Ex. Fis. Nat.

In this study, we investigated the extent to which adolescents who spend time playing violent video games exhibit higher levels of aggressive behaviour when compared with those who do not. A large sample of British adolescent participants ( n = 1004) aged 14 and 15 years and an equal number of their carers were interviewed. Young people provided reports of their recent gaming experiences. Further, the violent contents of these games were coded using official EU and US ratings, and carers provided evaluations of their adolescents' aggressive behaviours in the past month. Following a preregistered analysis plan, multiple regression analyses tested the hypothesis that recent violent game play is linearly and positively related to carer assessments of aggressive behaviour. Results did not support this prediction, nor did they support the idea that the relationship between these factors follows a nonlinear parabolic function. There was no evidence for a critical tipping point relating violent game engagement to aggressive behaviour. Sensitivity and exploratory analyses indicated these null effects extended across multiple operationalizations of violent game engagement and when the focus was on another behavioural outcome, namely, prosocial behaviour. The discussion presents an interpretation of this pattern of effects in terms of both the ongoing scientific and policy debates around violent video games, and emerging standards for robust evidence-based policy concerning young people's technology use.

The ring compression test was employed to investigate the relationship between Coulomb friction coefficient and Tresca friction factor in a Coulomb–Tresca friction model. The results indicated that the ratio ( k ) of Coulomb friction coefficient to Tresca friction factor varies under lubricated and dry conditions. The ratio k can be described by a parabolic function under lubricated conditions and by an exponential function under dry condition. There is a transitional region between lubricated and dry conditions.

The morphological characteristics of the canal section are significantly different from those of natural rivers. In this paper, the cross-sectional morphological characteristics of the canal are studied based on the measured section data of the inland waterway in Jiangsu Province, China. A regression analysis of the cross-section of the canal revealed that the shape of the stable section conformed to the parabolic function, and parameter A can be used to characterize the width of the section. The statistical analysis of the width-to-depth ratio of cross sections revealed that the width-to-depth ratio distribution range in those canals has obvious geographical differences. It is suggested that the quadratic function coefficient A and the width-to-depth ratio can be used as the two cross-section parameters to describe the cross-sectional shape of the canal waterway

Longitudinal momentum (${P}_{\mathrm{L}}$) distributions of projectilelike fragments produced at $E=290\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}/\mathrm{nucleon}$ are investigated. ${P}_{\mathrm{L}}$ distributions of fragments produced by Ar and Kr beams with a wide variety of targets (C, Al, Nb, Tb, and Au) were measured using the fragment separator at HIMAC. ${P}_{\mathrm{L}}$ distributions observed for fragments with a wide range of mass losses $\mathrm{\ensuremath{\Delta}}A$ (1--30 for Ar beam and 1--64 for Kr beam), show a slightly, but definitely asymmetric nature. The peak shift and width were obtained from the observed ${P}_{\mathrm{L}}$ distributions. No significant target dependence was found in either the peak shift or width. For the practical application, the variation in momentum peak shift with fragment mass (${A}_{\mathrm{F}}$) was represented by a parabolic function. The width on the high-${P}_{\mathrm{L}}$ side (${\ensuremath{\sigma}}_{\mathrm{High}}$) is well reproduced by the Goldhaber formula, which is obtained from the contribution of the Fermi momentum. The behavior of the reduced width, ${\ensuremath{\sigma}}_{0}$, obtained from ${\ensuremath{\sigma}}_{\mathrm{High}}$ via the Goldhaber formulation, is consistent with the mass-dependent Fermi momentum of a nucleon. The width on the low-${P}_{\mathrm{L}}$ side (${\ensuremath{\sigma}}_{\mathrm{Low}}$) is markedly larger than ${\ensuremath{\sigma}}_{\mathrm{High}}$ and exhibits a clear ${A}_{\mathrm{F}}$ dependence.

Clitellocephalus ophoni (Tuzet and Ormieres, 1956) Clopton, 2002, is one of the parasites of a common ground beetle species, Harpalus rufipes (De Geer, 1774), inhabiting practically the entire temperate zone of Eurasia. Photographs of 177 gamonts and 74 syzygies of C. ophoni from specimens of three populations of H. rufipes collected from the countryside near Dnipropetrovsk (Ukraine), were analyzed according to 15 linear characteristics and 18 indices for gamonts and 6 indices for syzygies. The coefficient of variation (CV) for the majority of linear parameters for gamonts ranged between 28.2-71.3%. Out of 15 linear parameters, only 6 showed a normal distribution; the others showed a statistically significant asymmetry (the maximum indicators on the histogram of distribution were inclined to the area of minimum value). The distribution of all 18 morphological indices diverged from the norm. Two maximum indicators were registered in the disposition of the widest point of the protomerite and deutomerite. The minimum CV was registered for the ratio of gamont length to the length of its deutomerite. The size ratios of the primite and satellite in the syzygies were more constant than the morphological indices for the gamonts. The sex of the host does not affect the length of the protomerite and deutomerite, but does affect their width, the disposition of the nucleus and the widest point of the deutomerite. The length of the protomerite and deutomerite relative to the length of the gamont is better described through linear functions, while their width has a nonlinear dynamic and is better described through parabolic function. The ratio of morphometric indices to total length of C. ophoni gamonts is also better described through nonlinear functions. The data obtained on the morphological variability of C. ophoni needs to be compared with the results for artificially infected individuals of other Harpalus species. Key words: Clitellocephalus ; Harpalus rufipes ; gregarine; morphometric characteristics; ratio of morphometric indices Received: September 23, 2015; Revised: November 30, 2015; Accepted: December 3, 2015; Published online: May 10, 2016

We investigate the characteristics of Galactic cosmic rays in events associated with magnetic clouds that reach Earth. A mathematical model, capable of describing the distribution of the cosmic-ray density in a magnetic cloud is considered. We show that in most cases the behavior of the cosmic-ray density within magnetic clouds at 1 AU can be described accurately by a parabolic function of the distance to the center of the magnetic cloud measured in gyroradii. As expected, the majority of magnetic clouds modulate cosmic rays, resulting in a reduction of their density. However, there is a group of events (about one fifth of the total sample) in which the density of cosmic rays in a magnetic cloud increases. Furthermore, the extremum (a minimum or a maximum) of the cosmic-ray density is found closer to the cloud center and not at its edges. We consider a number of the factors contributing to the model and estimate the effect of each factor.

This paper proposes a system of nonlinear equations to manipulate the amplitude of parabolic function of transmission errors. Firstly, the characteristics of parabolic function of transmission errors are defined. Then, a system of nonlinear equations for manipulating the amplitude of parabolic function of transmission errors is created based on both the conditions of contact and the constraint on the amplitude of function of transmission errors. As the number of independent scalar equations in the system minus the number of unknown parameters is one, one extra design parameter can be applied to manipulate the amplitude of parabolic function of transmission errors. The solution to the extra design variable is automatically, precisely, and efficiently determined by the computer program which is created based on the Newton’s root finding method. The time-consuming manual iterations for trying the value of design variable are eliminated. The proposed method can be applied to both two-and three-dimensional gearing problems. At last, a pair of meshing gears composed of a circular-arc spur gear and an involute spur gear is presented to verify the methodology proposed in this paper.

We study a class of nonlocal-diffusion equations with drifts, and derive a priori \(\Phi \)-Holder estimate for the solutions by using a purely probabilistic argument, where \(\Phi \) is an intrinsic scaling function for the equation.

This research is performed to evaluate the divergence angle of far-field distribution of the collimation system. A concise method that can analyze the divergence performance of a collimation system is put forward. Simulation methods for beam truncated systems and beam untruncated systems are proposed. For truncated beams, a proper parabolic distribution at z = 0 is used to approximate the beam with “bell-shaped” EFD (equivalent field distribution). Results show that the flat-top beam ( h = 1 ) has the minimal divergence angle. The parabolic function is easier to process than the Gaussian function. For untruncated beams, the calculation results indicate that the Lorentzian beam has a minimum divergence angle. It is foreseeable that this study will play a positive role in driving the design and evaluation of the collimation system.

Objective The objective of this study was to use two-dimensional (2D) ultrasound (US) during routine prenatal surveillance to develop normative estimated placental volume (EPV) growth curves. Study Design Patients ≥ 18 years old with singleton pregnancies were prospectively followed from 11 weeks gestational age (GA) until delivery. At routine US visits, placental width, height, and thickness were measured and EPV calculated using a validated mathematical model. Results In this study, 423 patients were scanned between 9.7 and 39.3 weeks GA to generate a total of 627 EPV calculations. Readings were clustered at 12 and 20 weeks, times of routine scanning. The mean EPV was 73 ± 47 cc at 12.5 ± 1.5 weeks (n = 444) and 276 ± 106 cc at 20 ± 2 weeks (n = 151). The data best fit a parabolic function as follows: EPV = (0.384GA – 0.00366GA2)3. Tenth and 90th percentile lines were generated with ± 1.28 SE offset. EPV readings below the 10th or above the 90th percentiles tended to be associated with either small or large newborns, respectively. Conclusion Routine 2D US created EPV growth curves, which may be useful for stratifying patients into prenatal risk groups.

This paper presents a corner-based image alignment algorithm based on the procedures of corner-based template matching and geometric parameter estimation. This algorithm consists of two stages: 1) training phase, and 2) matching phase. In the training phase, a corner detection algorithm is used to extract the corners. These corners are then used to build the pyramid images. In the matching phase, the corners are obtained using the same corner detection algorithm. The similarity measure is then determined by the differences of gradient vector between the corners obtained in the template image and the inspection image, respectively. A parabolic function is further applied to evaluate the geometric relationship between the template and the inspection images. Results show that the corner-based template matching outperforms the original edge-based template matching in efficiency, and both of them are robust against non-liner light changes. The accuracy and precision of the corner-based image alignment are competitive to that of edge-based image alignment under the same environment. In practice, the proposed algorithm demonstrates its precision, efficiency and robustness in image alignment for real world applications.

This study deals with the analysis of relationship between 67 environmental indicators in geological environment and breast cancer mortality in Slovakia. Primary data comprises a database of 67 environmental indicators for groundwater and soils as a mean value for every Slovak municipality and the data on relative mortality from breast cancer for Slovak municipalities (about 2900) collected for the period of 22 years. The training neural network was used as a mathematical model for data analysis. The top 200 networks have identified 12 environmental indicators (8 for soils: pHKCl, carbonates, Ca, Mg, Na, Co, Sr and 4 for groundwater: Ba, SiO2, Zn, Fe) with high rate of influence. As the most influential environmental indicator was identified pHKCl in soil that should be within the range of acids, definitely under the neutral point. Most of environmental indicators were unambiguously identified as negative (Mg, Ca, Sr, carbonates, Ba, SiO2, Zn and Fe). These indicators should be at the lowest level of their concentration in related geological component. Three environmental indicators, Ce, Na and Co had parabolic function where certain essential concentration levels of these elements are presumed with protective effect on human health. For all influential indicators limit values with respect to the lowest mortality were proposed. The verification of achieved results was performed through regressive model of breast cancer mortality for 12 calculated influential environmental indicators for all municipalities in the Slovak Republic. The revealed difference between real mortality levels and the model values was found for 6 environmental indicators (pHKCl, Mg, Ca, Sr, Ba, Zn). Each executed limit has decreased breast cancer mortality for about 0.79 cases per 100,000 inhabitants.

The refractive index distribution of LC GRIN lenses for autostereoscopic 3D displays was investigated. It was clarified that the refractive index distribution forms a parabolic function of the optimum lens conditions. We achieved a minimum crosstalk of 0.6% in an autostereoscopic 3D display with an LC GRIN lens.

The study was carried out in Serbian spruce (Picea omorika Panc/Purkyn?) plantations in the western Serbia. The paper presents results of the analysis of crown development. The following elements were analyzed: total tree height, height of the crown base, absolute and relative crown length, maximal crown diameter, coefficient of crown spreading and degree of crown girth. We discuss approaches to the modeling of tree crown growth and development, growing under favorable environmental and stand conditions, without anomalies in development. In order to establish the relationship between analyzed factors, regression analyses were applied. Data fitting was by the analytic method, by the implementation of Prodan?s functions of growth, linear and parabolic function. Received models can be used for the simulation of various growth and developing processes in forest.

In this paper, the comparative study of RWIV will be proceeding. Aim at the model of rivulet, a kind of sine function is accepted, and another kind of rivulet model is established on the basis of analysis on the stress of the rivulet. Aim at the model of cable’s initial curve under gravity, a kind of parabolic function is accepted, and another kind of model accepted as a catenary. Then, solve the equations of cable and the rivulet jointly by numerical method, get the characteristics of motion of the cable and rivulet, then investigate the factors which is related with the model. The influence of the different model of cable is very little when the rivulet model is assumed as a sine function. However, the influence of the different model of cable is large when the rivulet model is established on the basis of analysis on the stress. Then discover the catenary cable with moving rivulet is more rational with the field observation.

DPIV (Digital Particle Image Velocimetry) is often used to record the flows in silo models and to recognise the flow in two-dimensional structures. Flow in a two-dimensional laboratory model with transparent walls was recorded using the DPIV technique. Recorded images were processed and the velocities of the flowing material were obtained. Statistical analysis of the readings taken from velocity profiles was also performed. To describe the flow the theoretical model of velocity presented in Choi, Kudrolli, and Bazant (2005) was modified. An empirical description of the flow rate was carried out. An analysis of the kinematic parameter b was carried out and compared to the solutions given by Medina, Cordova, Luna, and Trevino (1998) and Choi et al. (2005) . Comparisons between the experimental measurements and the empirical descriptions are presented. Comparing the Gaussian based kinematic model of Choi et al. (2005) and a modified empirical kinematic model, a better description of velocity was obtained by using the latter. In further comparisons using the two aformentioned models and a parabolic description the best description of velocity was given by the parabolic function. Flow rate was analysed using the three types of functions. Both the Gaussian and modified empirical kinematic model gave almost the same values for calculated levels h = 5, 10 cm. Using the parabolic description the value of flow rates differed slightly. The flow channel boundary was analysed using parabolic and hyperbolic descriptions. Both descriptions were good because the correlation coefficients, had values ranging from 0.928 to 0.997.

Power-angle curves, describing the isotonic sub-maximal movement of the lower extremities, are better means to evaluate the effects of training protocols and to highlight functional differences between the legs. The aim of this work was to compare the morphology of power-angle curves of pathological subjects (PS) and healthy subjects (HS) and to evaluate the feasibility of this approach to discriminate between pathological and healthy legs. Isotonic evaluation with leg-extension machine was performed on 46 HS and 16 PS previously submitted to knee surgery. The healthy group comprised 33 elderly healthy subjects (EHS) (mean: 65 years) and 13 young healthy subjects (YHS) (mean: 35 years) and the influence of age on the curves was evaluated. All subjects performed a sub-maximal movement with the legs working alone and together. Each curve was interpolated with a parabolic function and the R 2 Pearson coefficient was used as a score for morphological regularity. R 2 of curves of PS obtained with legs working alone were statistically different from those of the HS group, considering both YHS and EHS (p < 0.05). Statistically significant differences between YHS and EHS (p < 0.01) show the importance of age in the modality of performing movements.

Recently, there is a growing interest in two-dimensional (2D) plane indentation as an imprinting method for creating nanostructures. There is also a strong interest in using 2D flat-ended, wedge, and cylindrical indenters for characterizing mechanical properties of materials. In either case the knowledge of load versus displacement of the indenter is important. However, there has been some confusion about the load–displacement relationships for 2D indentation in the literature. Concerning this confusion on the relationship between the indentation load and the indentation depth for 2D elastic indentations, the symmetric indentation of an elastic half-space is studied. Parameters are introduced in determining the semianalytical relation between the indentation load and the indentation depth for flat-ended indenters and in determining the dependence of the indentation depth on the contact size for non-flat-ended indenters. The indentation load is proportional to the indentation depth for the indentation by flat-ended indenters and is a parabolic function of the indentation depth to the first order of approximation for non-flat-ended indenters including the wedge and cylindrical indenters.

Commercial samples of a white zirconium frit and a Cr–Fe–Al–Zn pigment were used as a diffusion pair. Interdiffusion was observed at temperatures >1000 °C, indicating that the growth of the formed interface can be described by a parabolic function, which is dependent on the diffusion coefficients of the reacting elements, the chemical potential of these elements and the system geometry.

Highway infrastructure elements, such as signs, guardrails, structures, and pavements, deteriorate over time because of weather, accidents, daily wear and tear, and other factors. To keep them functional and operational, the elements require periodic maintenance. In the absence of an optimization model that can optimally allocate the annual maintenance budget for needed maintenance, it is often difficult to secure adequate funding for such activities. A mathematical model is developed; it evaluates the condition of elements, considers a budget constraint, and suggests the optimal maintenance schedule over a specified period of time. A genetic algorithm has been used to solve the model. The model considers a parabolic function over time to represent deterioration of the elements as well as rehabilitation expenses. The extent of rehabilitation at a given time is considered as dependent on the present condition and amount of deterioration. Also, threshold values for level of deterioration are considered as a check for the minimum condition before rehabilitation is warranted. With inputs such as deterioration levels, rehabilitation costs, threshold deterioration levels, and annual budget, the model can give the optimum maintenance schedule of the elements over a specified time period as an output. The approach is illustrated through a numerical example.

The analytical solution for the plane contact between nonconforming C1 profiles with a curvature discontinuity is obtained and discussed. The initial gap between the undeformed profiles was represented by a double parabolic function having the curvature discontinuity within the contact region. On the basis of the analytical solution, the complete analysis of pressure distribution was obtained. A raw technique to find an approximate solution is presented and comparison to the analytic one provided. For the limit case of strong curvature ratio the asymptotic treatment is discussed in terms of consistency with hypotheses claimed.

The primary focus of this paper was to develop a high-performance computer system for optimizing auditory stimuli based on neuronal feedback. Using the Algorithm Of Pattern EXtraction (ALOPEX) extra-cellular action potentials (APs) recorded from frog (Rana Pipiens) auditory neurons were used as feedback to optimize sound stimuli. This computer-based system works in real time to iteratively find the neuron's best excitatory frequency (BEF). Three programmable (positive and negative) threshold logic levels are used to collect 300 APs in response to normalized pure tones. Fuzzy logic is then used to separate up to five fuzzy centers (templates) from the 300 APs. The fuzzy centers are used for on-line fuzzy mapping of future responses. The five fuzzy centers allow the system to monitor up to five neighboring neurons. To study the auditory neurons of the frog, one, two, and three simultaneous tones are used as the stimulus for optimization of the best combination of frequencies. Testing with the response calculated as a parabolic function of a single best frequency demonstrated system dynamics and reliability for up to nine simultaneous tones. Experiments using one pure tone and ten stimulus presentations per iteration showed that the automated system is able to repeatedly converge to the best frequency within 100 iterations. Studies using one, two, and then three pure tones played simultaneously on the same group of neurons has shown that these tones converged on the same best frequencies by properly mixing the tones available to produce the optimal complex sound.

On the basis of the paraxial wave equation of ultrashort pulsed beams, the free-space propagation equation of the space-time Wigner distribution function has been derived. The propagation laws for the moments of ultrashort pulsed beams are given. A class of propagation-invariant moments is discovered. The second-order temporal central moment is a parabolic function of the propagation distance and the root-mean-square width of ultrashort pulses increases during the propagation.