Quasiconformal mappings and inequalities involving special functions

This PhD thesis in Mathematics belongs to the field of Geometric Function Theory. The thesis consists of four original papers. The topic studied deals with quasiconformal mappings and their distortion theory in Euclidean n-dimensional spaces. This theory has its roots in the pioneering papers of F. W. Gehring and J. Väisälä published in the early 1960’s and it has been studied by many mathematicians thereafter. In the first paper we refine the known bounds for the so-called Mori constant and also estimate the distortion in the hyperbolic metric. The second paper deals with radial functions which are simple examples of quasiconformal mappings. These radial functions lead us to the study of the so-called p-angular distance which has been studied recently e.g. by L. Maligranda and S. Dragomir. In the third paper we study a class of functions of a real variable studied by P. Lindqvist in an influential paper. This leads one to study parametrized analogues of classical trigonometric and hyperbolic functions which for the parameter value p = 2 coincide with the classical functions. Gaussian hypergeometric functions have an important role in the study of these special functions. Several new inequalities and identities involving p-analogues of these functions are also given. In the fourth paper we study the generalized complete elliptic integrals, modular functions and some related functions. We find the upper and lower bounds of these functions, and those bounds are given in a simple form. This theory has a long history which goes back two centuries and includes names such as A. M. Legendre, C. Jacobi, C. F. Gauss. Modular functions also occur in the study of quasiconformal mappings. Conformal invariants, such as the modulus of a curve family, are often applied in quasiconformal mapping theory. The invariants can be sometimes expressed in terms of special conformal mappings. This fact explains why special functions often occur in this theory.

Vegetative propagation of bauhinia x blakeana, an ornamental sterile tree

Bauhinia x blakeana (B. purpurea x B. variegata) is a natural hybrid that has been cultivated in gardens, streets and parks. Due to its sterility, it must be vegetatively propagated. The objective of this work was to evaluate the viability of cuttings and grafting on its propagation. Semi-woody cuttings were collected during four seasons and treated with 0; 1,000; 2,000; and 3,000 mg L-1 of IBA. The experimental design was entirely randomized and the treatments were arranged in a 4x4 factorial scheme (four collecting times x four IBA concentrations) and five replications with 10 cuttings each, per collecting time and per IBA concentration. Characteristics of roots and shoots were evaluated after 90 days. The data means were compared by the Tukey test and submitted to the polynomial regression analysis. For the grafting experiment, B. variegata and B. variegata var. candida plants of six and 12 months were used as rootstocks and the splice graft and T-budding methods were tested. The experimental design was entirely randomized and the treatments were arranged in a 2x2x2 factorial scheme (two rootstock species x two grafting methods x two rootstock ages) and four replications with five plants each, per rootstock species, per grafting method and per rootstock age. Characteristics of shoots were evaluated after 90 days and the data means were compared by the Tukey test. B. x blakeana can be propagated by semi-woody cuttings collected in spring, without IBA application, or in summer, with the application of 3,000 mg L-1 of IBA. The tested grafting methods were not effective.

Development of capacitive sensor for measuring soil water content

The irrigation management based on the monitoring of the soil water content allows for the minimization of the amount of water applied, making its use more efficient. Taking into account these aspects, in this work, a sensor for measuring the soil water content was developed to allow real time automation of irrigation systems. This way, problems affecting crop yielding such as irregularities in the time to turn on or turn off the pump, and excess or deficit of water can be solved. To develop the sensors were used stainless steel rods, resin, and insulating varnish. The sensors measuring circuit was based on a microcontroller, which gives its output signal in the digital format. The sensors were calibrated using soil of the type “Quartzarenic Neosoil”. A third order polynomial model was fitted to the experimental data between the values of water content corresponding to the field capacity and the wilting point to correlate the soil water content obtained by the oven standard method with those measured by the electronic circuit, with a coefficient of determination of 93.17%, and an accuracy in the measures of ±0.010 kg kg-1. Based on the results, it was concluded that the sensor and its implemented measuring circuit can be used in the automation process of irrigation systems.

Estimation of transpiration of the 'Valencia' orange young plant using thermal dissipation probe method

Most studies on measures of transpiration of plants, especially woody fruit, relies on methods of heat supply in the trunk. This study aimed to calibrate the Thermal Dissipation Probe Method (TDP) to estimate the transpiration, study the effects of natural thermal gradients and determine the relation between outside diameter and area of xylem in 'Valencia' orange young plants. TDP were installed in 40 orange plants of 15 months old, planted in boxes of 500 L, in a greenhouse. It was tested the correction of the natural thermal differences (DTN) for the estimation based on two unheated probes. The area of the conductive section was related to the outside diameter of the stem by means of polynomial regression. The equation for estimation of sap flow was calibrated having as standard lysimeter measures of a representative plant. The angular coefficient of the equation for estimating sap flow was adjusted by minimizing the absolute deviation between the sap flow and daily transpiration measured by lysimeter. Based on these results, it was concluded that the method of TDP, adjusting the original calibration and correction of the DTN, was effective in transpiration assessment.

Longitudinal distribution of bean seeds in horizontal plate meter operating with one or two seed outlets

This study aimed to evaluate the effect of horizontal plate meter with oblong holes operating with one or two seed outlets at different speeds over the accuracy of longitudinal distribution of common bean seeds (Phaseolus vulgaris L.). The experiment was performed in laboratory using the meter with one and two seed outlet points in relation to ten tangential disk plate speeds (0.03 to 0.30 m s-1). It was used a complete randomized design with four replications, summing up 80 experimental treatments. Tangential speed quantitative factor was estimated through a 2nd order polynomial regression. There was no significant difference in the behavior of the seed meter operating with one or two outlets in the metering of bean seeds in all tested speeds, with percentage of single seeds spacing over 60% in tangential speeds below of 0.24 m s-1.

Economic analysis of cashew early dwarf Crop BRS - 189 depending on water levels and doses of potassium fertilization

Water and fertilizer among the production factors are the elements that most restrict the production of cashew. The precise amount of these factors is essential to the success of the crop yield. This research aimed to determine the best factor-product ratio and analyze technical and economic indicators, of productivity of the cashew clone BRS 189 (Anacardium occidentale) to production factors water and potassium. The experiment was conducted from May 2009 to December 2009 in an experimental area of 56.0 m x 112.0 m in the irrigated Curu - Pentecoste, located in the municipality of Pentecoste, Ceará, Brazil. Production factors water (W) and potassium (K) were the independent variables and productivity (Y), the dependent variable. Ten statistical models that have proven satisfactory for obtaining production function were tested. The marginal rate of substitution was obtained through the ratio of the potassium marginal physical product and the water marginal physical product. The most suited model to the conditions of the experiment was the quadratic polynomial without intercept and interaction. Considering that the price of the water was 0.10 R$ mm -1, the price of the potassium 2.19 R$ kg -1 and the price of the cashew 0.60 R$ kg-1, the amounts of water and K2O to obtain the maximum net income were 6,349.1 L plant-1 of water and 128.7 g plant -1year, -1 respectively. Substituting the values obtained in the production function, the maximum net income was achieved with a yield of 7,496.8 kg ha-1 of cashew.

Effects of starter diet supplementation with arginine on broiler production performance and on small intestine morphometry

The effects of starter diet (days 1 to 21) supplemented with arginine (Arg) on the production performance and duodenum and jejunum mucosa morphometry of broilers were studied. Male Cobb broiler chickens (990) were randomly assigned to one of five treatments in a complete random design. Measurements of 33 chicks per treatment were made in six repetitions. The treatments consisted of a basal diet with 1.390% digestible Arg (no supplementation) and four dietary levels (1.490%, 1.590%, 1.690%, and 1.790%), providing a relationship with lysine of 1.103; 1.183; 1.262; 1.341 and 1.421%, respectively. From the age of 22 days on, all birds received conventional grower diet. The data were submitted to regression analysis by polynomial decomposition of the degrees of freedom in relation to the levels of Arg. The Arg supplementation increased (P<0.05) the live weight and the feed conversion ratio without increasing the feed intake of the birds. However, no effect was observed (P>0.05) in the growth phase (days 22 to 42) in the absence of the Arg supplementation. The supplementation of Arg over of NRC recommendation during the starter phase may be necessary for the expression of the maximal weight gain potential in birds. No effect (P<0.05) of Arg dietary supplementation was observed either on small intestine weight and length at any age. However, the duodenum villus:crypt ratio increased and the crypt depth decreased in the first week in response to increasing dietary Arg. It is concluded that broiler Arg dietary supplementation in the starter diet improved production performance and small intestine morphometry, especially in the first week.

Control of Structures with Cubic and Quadratic Non-Linearities with Time Delay Consideration

This paper studies the effect of time delay on the active non-linear control of dynamically loaded flexible structures. The behavior of non-linear systems under state feedback control, considering a fixed time delay for the control force, is investigated. A control method based on non-linear optimal control, using a tensorial formulation and state feedback control is used. The state equations and the control forces are expressed in polynomial form and a performance index, quadratic in both state vector and control forces, is used. General polynomial representations of the non-linear control law are obtained and implemented for control algorithms up to the fifth order. This methodology is applied to systems with quadratic and cubic non-linearities. Strongly non-linear systems are tested and the effectiveness of the control system including a delay for the application of control forces is discussed. Numerical results indicate that the adopted control algorithm can be efficient for non-linear systems, chiefly in the presence of strong non-linearities but increasing time delay reduces the efficiency of the control system. Numerical results emphasize the importance of considering time delay in the project of active structural control systems.

Substitution-Newton-Raphson method applied to the modeling of a vapour compression refrigeration system using different representations of the thermodynamic properties of R-134a

This paper gives a detailed presentation of the Substitution-Newton-Raphson method, suitable for large sparse non-linear systems. It combines the Successive Substitution method and the Newton-Raphson method in such way as to take the best advantages of both, keeping the convergence features of the Newton-Raphson with the low requirements of memory and time of the Successive Substitution schemes. The large system is solved employing few effective variables, using the greatest possible part of the model equations in substitution fashion to fix the remaining variables, but maintaining the convergence characteristics of the Newton-Raphson. The methodology is exemplified through a simple algebraic system, and applied to a simple thermodynamic, mechanical and heat transfer modeling of a single-stage vapor compression refrigeration system. Three distinct approaches for reproducing the thermodynamic properties of the refrigerant R-134a are compared: the linear interpolation from tabulated data, the use of polynomial fitted curves and the use of functions derived from the Helmholtz free energy.

Welding current effect on diffusible hydrogen content in flux cored arc weld metal

The application of flux cored arc welding (FCAW) has increased in manufacturing and fabrication. Even though FCAW is well known for its good capability in producing quality welds, few reports have been published on the cause of the relatively high diffusible hydrogen content in the weld metal and its relation with the ingredients used in the wire production and with the welding parameters (mainly welding current). This paper describes experiments where data obtained from weld metal diffusible hydrogen analysis, metal droplet collection, and high-speed recording of metal droplet transfer were used to evaluate the effect of welding current on diffusible hydrogen content in the weld metal. The results from gas chromatography analysis showed that weld metal hydrogen content indeed increased with welding current. A polynomial regressional analysis concluded that hydrogen increase with current was better described by a linear function with proportional constant of approximately 0.7 or 70%. Different from the GMA welding transfer behavior, statistical analysis showed only a small increase in metal droplet size with increasing current. The metal transfer mode remained in the globular range for currents between 100 and 150 A. The most surprising findings were with the high-speed cinematography recording. Observing the high speed movies, it was possible to see that at low current, "unmelted" flux sporadically touched the weld pool but at higher current, the flux remained touching the weld pool during the whole time of droplet formation and transfer. It is believed that since the flux has ingredients that contain hydrogen, hydrogen passes through the arc undisturbed, going to the weld bead intact and increasing the hydrogen content in the weld metal. Another important observation is regarding to droplet size. Droplet size increased with increasing current because forces from decomposed gases from the flux could sustain the droplets, retarding their transfer and allowing them to grow.

Adaptive control of Hydraulic Drive

Adaptive control systems are one of the most significant research directions of modern control theory. It is well known that every mechanical appliance’s behavior noticeably depends on environmental changes, functioning-mode parameter changes and changes in technical characteristics of internal functional devices. An adaptive controller involved in control process allows reducing an influence of such changes. In spite of this such type of control methods is applied seldom due to specifics of a controller designing. The work presented in this paper shows the design process of the adaptive controller built by Lyapunov’s function method for the Hydraulic Drive. The calculation needed and the modeling were conducting with MATLAB® software including Simulink® and Symbolic Math Toolbox™ etc. In the work there was applied the Jacobi matrix linearization of the object’s mathematical model and derivation of the suitable reference models based on Newton’s characteristic polynomial. The intelligent adaptive to nonlinearities algorithm for solving Lyapunov’s equation was developed. Developed algorithm works properly but considered plant is not met requirement of functioning with. The results showed confirmation that adaptive systems application significantly increases possibilities in use devices and might be used for correction a system’s behavior dynamics.

A comparative analysis of preprocessing techniques of cardiac event series for the study of heart rhythm variability using simulated signals

In the present study, using noise-free simulated signals, we performed a comparative examination of several preprocessing techniques that are used to transform the cardiac event series in a regularly sampled time series, appropriate for spectral analysis of heart rhythm variability (HRV). First, a group of noise-free simulated point event series, which represents a time series of heartbeats, was generated by an integral pulse frequency modulation model. In order to evaluate the performance of the preprocessing methods, the differences between the spectra of the preprocessed simulated signals and the true spectrum (spectrum of the model input modulating signals) were surveyed by visual analysis and by contrasting merit indices. It is desired that estimated spectra match the true spectrum as close as possible, showing a minimum of harmonic components and other artifacts. The merit indices proposed to quantify these mismatches were the leakage rate, defined as a measure of leakage components (located outside some narrow windows centered at frequencies of model input modulating signals) with respect to the whole spectral components, and the numbers of leakage components with amplitudes greater than 1%, 5% and 10% of the total spectral components. Our data, obtained from a noise-free simulation, indicate that the utilization of heart rate values instead of heart period values in the derivation of signals representative of heart rhythm results in more accurate spectra. Furthermore, our data support the efficiency of the widely used preprocessing technique based on the convolution of inverse interval function values with a rectangular window, and suggest the preprocessing technique based on a cubic polynomial interpolation of inverse interval function values and succeeding spectral analysis as another efficient and fast method for the analysis of HRV signals

Antioxidant activity of rosemary and oregano ethanol extracts in soybean oil under thermal oxidation

Four experiments were conducted to measure the antioxidant activity of ethanol extracts of rosemary and oregano compared with synthetic antioxidants such as TBHQ and BHA/BHT. The antioxidant activity was determined and results differed from those of the Oven test at 63º C. Peroxide values and absorptivities at 232 nm of soybean oil under Oven test were lower in treatments with 25, 50, 75, 100 and 200 mg.Kg-1 TBHQ than in treatments with 1000 mg.Kg-1 oregano extract (O), 500 mg.Kg-1 rosemary extract (R) and their mixture R+O. All the treatments were effective in controlling the thermal oxidation of oils; the natural extracts were as effective as BHA+BHT and less effective than TBHQ. The natural extracts were mixed with 25, 50, 75 and 100 mg.Kg-1 TBHQ and then added to the oil. No improvement in antioxidative properties was observed. The best antioxidant concentration could be determined from polynomial regression and quadratic equation from the experimental data.

Shirshovin lause PI-algebroissa

Tässä työssä esitetään venäläisen matemaatikon A.I. Shirshovin teorioita ja tuloksia sanojen kombinatoriikasta. Lisäksi näytetään miten ne soveltuvat PI-algebrojen maailmaan. Shirshovin tuloksia tarkasteltaessa käsitellään sanoja erillisinä kombinatorisina objekteina ja todistetaan Shirshovin Lemma, joka on tämän työn perusta. Lemmanmukaan tarpeeksi pitkille sanoille saadaan tiettyä säännönmukaisuutta ja se todistetaan kolme kertaa. Ensimmäisestä saadaan tarpeeksi pitkän sanan olemassaolo.Toinen todistus mukailee Shirshovin alkuperäistä todistusta. Kolmannessa todistuksessa annetaan tarpeeksi pitkälle sanalle paremmin käytäntöön soveltuva raja. Tämän jälkeen käsitellään sanoja algebrallisina objekteina. Työn päätuloksena todistetaan Shirshovin Korkeuslause, jonka mukaan jokainen äärellisesti generoidunPI-algebran alkio on sanojen ω1k1 ···ωdkd lineaarikombinaatio, missä sanojen ωi pi-tuudet sekä indeksi i ovat rajatut. Shirshovin Korkeuslauseesta seuraa suoraan positiivinen ratkaisu Kurochin ongelmaan PI-algebroilla sekä saadaan raja alkioiden lukumäärälle, jolla algebra generoituu moduliksi. Lisäksi esitetään toisena sovelluksena ilman todistuksia Shirshovin soveltuvuus Jacobsonin radikaalin nilpotenttisuuteen. Pääsääntöisenä lähteenä käytetään A. Kanel-Belowin ja L. H. Rowenin kirjaa: Computational aspects of polynomial identities.

Rheological and thermophysical properties of blackberry juice

Rheological and thermophysical properties were determined for blackberry juice, which was produced from blackberry fruit at 9.1 ± 0.5 °Brix and density of 1.0334 ± 0.0043 g cm-3. The concentration process was performed using a roto evaporator, under vacuum, to obtain concentrated juice at about 60 °Brix. In order to obtain different concentrations, concentrated juice was diluted with distilled water. Rheological measurements were carried out using a Rheotest 2.1 Searle type rheometer. In the tested ranges, the samples behaved as pseudoplastic fluids, and the Power-Law model was satisfactorily fitted to the experimental data. The friction factor was measured for blackberry juice in laminar flow conditions of pseudoplastic behavior. Thermal conductivity, thermal diffusivity and density of blackberry juice at 9.4 to 58.4 °Brix were determined, in triplicate, at 0.5 to 80.8 °C. Polynomial regression was performed to fit experimental data obtaining a good fit. Both temperature and concentration showed a strong influence on thermophysical properties of blackberry juice. Calculated apparent specific heat values varied from 2.416 to 4.300 kJ.kg-1 °C in the studied interval.