Fractional derivatives: probability interpretation and frequency response of rational approximations

The theory of fractional calculus (FC) is a useful mathematical tool in many applied sciences. Nevertheless, only in the last decades researchers were motivated for the adoption of the FC concepts. There are several reasons for this state of affairs, namely the co-existence of different definitions and interpretations, and the necessity of approximation methods for the real time calculation of fractional derivatives (FDs). In a first part, this paper introduces a probabilistic interpretation of the fractional derivative based on the Grünwald-Letnikov definition. In a second part, the calculation of fractional derivatives through Padé fraction approximations is analyzed. It is observed that the probabilistic interpretation and the frequency response of fraction approximations of FDs reveal a clear correlation between both concepts.

Fractional-order impulse response of the respiratory system

This paper presents the measurement, frequency-response modeling and identification, and the corresponding impulse time response of the human respiratory impedance and admittance. The investigated adult patient groups were healthy, diagnosed with chronic obstructive pulmonary disease and kyphoscoliosis, respectively. The investigated children patient groups were healthy, diagnosed with asthma and cystic fibrosis, respectively. Fractional order (FO) models are identified on the measured impedance to quantify the respiratory mechanical properties. Two methods are presented for obtaining and simulating the time-domain impulse response from FO models of the respiratory admittance: (i) the classical pole-zero interpolation proposed by Oustaloup in the early 90s, and (ii) the inverse discrete Fourier Transform (DFT). The results of the identified FO models for the respiratory admittance are presented by means of their average values for each group of patients. Consequently, the impulse time response calculated from the frequency response of the averaged FO models is given by means of the two methods mentioned above. Our results indicate that both methods provide similar impulse response data. However, we suggest that the inverse DFT is a more suitable alternative to the high order transfer functions obtained using the classical Oustaloup filter. Additionally, a power law model is fitted on the impulse response data, emphasizing the intrinsic fractal dynamics of the respiratory system.

Modelização de equivalente eléctrico para rede de nervuras de folhas vegetais

A tese tem como objectivo principal a criação de um modelo equivalente eléctrico da rede de nervuras de algumas folhas vegetais e analisar o seu comportamento a estímulos eléctricos, analisando-se também a respectiva resposta em frequência. A motivação desta tese passa pela observação dos sistemas existentes na natureza. Neste caso, as folhas vegetais e analisar se são sistemas de ordem fraccionária ou não. Para a sua elaboração, fez-se uma breve abordagem à estrutura das plantas, sob o ponto de vista da botânica e elaborou-se um método de fotografia das amostras, mapeamento da rede de nervuras e medição dos segmentos que compõem essa mesma rede. A tese termina com um capítulo de resultados experimentais.

On the analysis of data emerging in non-linear and complex systemscomparison of two X-Ray prony spectra

New arguments proving that successive (repeated) measurements have a memory and actually remember each other are presented. The recognition of this peculiarity can change essentially the existing paradigm associated with conventional observation in behavior of different complex systems and lead towards the application of an intermediate model (IM). This IM can provide a very accurate fit of the measured data in terms of the Prony's decomposition. This decomposition, in turn, contains a small set of the fitting parameters relatively to the number of initial data points and allows comparing the measured data in cases where the “best fit” model based on some specific physical principles is absent. As an example, we consider two X-ray diffractometers (defined in paper as A- (“cheap”) and B- (“expensive”) that are used after their proper calibration for the measuring of the same substance (corundum a-Al2O3). The amplitude-frequency response (AFR) obtained in the frame of the Prony's decomposition can be used for comparison of the spectra recorded from (A) and (B) - X-ray diffractometers (XRDs) for calibration and other practical purposes. We prove also that the Fourier decomposition can be adapted to “ideal” experiment without memory while the Prony's decomposition corresponds to real measurement and can be fitted in the frame of the IM in this case. New statistical parameters describing the properties of experimental equipment (irrespective to their internal “filling”) are found. The suggested approach is rather general and can be used for calibration and comparison of different complex dynamical systems in practical purposes.

Modeling of the lung impedance using a fractional order ladder network with constant phase elements

The self similar branching arrangement of the airways makes the respiratory system an ideal candidate for the application of fractional calculus theory. The fractal geometry is typically characterized by a recurrent structure. This study investigates the identification of a model for the respiratory tree by means of its electrical equivalent based on intrinsic morphology. Measurements were obtained from seven volunteers, in terms of their respiratory impedance by means of its complex representation for frequencies below 5 Hz. A parametric modeling is then applied to the complex valued data points. Since at low-frequency range the inertance is negligible, each airway branch is modeled by using gamma cell resistance and capacitance, the latter having a fractional-order constant phase element (CPE), which is identified from measurements. In addition, the complex impedance is also approximated by means of a model consisting of a lumped series resistance and a lumped fractional-order capacitance. The results reveal that both models characterize the data well, whereas the averaged CPE values are supraunitary and subunitary for the ladder network and the lumped model, respectively.

Development of fractional order capacitors based on electrolyte processes

In recent years, significant research in the field of electrochemistry was developed. The performance of electrical devices, depending on the processes of the electrolytes, was described and the physical origin of each parameter was established. However, the influence of the irregularity of the electrodes was not a subject of study and only recently this problem became relevant in the viewpoint of fractional calculus. This paper describes an electrolytic process in the perspective of fractional order capacitors. In this line of thought, are developed several experiments for measuring the electrical impedance of the devices. The results are analyzed through the frequency response, revealing capacitances of fractional order that can constitute an alternative to the classical integer order elements. Fractional order electric circuits are used to model and study the performance of the electrolyte processes.

Robot Manipulator Systems: Analysis and Control

Manipulator systems are rather complex and highly nonlinear which makes difficult their analysis and control. Classic system theory is veil known, however it is inadequate in the presence of strong nonlinear dynamics. Nonlinear controllers produce good results [1] and work has been done e. g. relating the manipulator nonlinear dynamics with frequency response [2–5]. Nevertheless, given the complexity of the problem, systematic methods which permit to draw conclusions about stability, imperfect modelling effects, compensation requirements, etc. are still lacking. In section 2 we start by analysing the variation of the poles and zeros of the descriptive transfer functions of a robot manipulator in order to motivate the development of more robust (and computationally efficient) control algorithms. Based on this analysis a new multirate controller which is an improvement of the well known “computed torque controller” [6] is announced in section 3. Some research in this area was done by Neuman [7,8] showing tbat better robustness is possible if the basic controller structure is modified. The present study stems from those ideas, and attempts to give a systematic treatment, which results in easy to use standard engineering tools. Finally, in section 4 conclusions are presented.

Influence of wearing an unstable shoe on thigh and leg muscle activity and venous response in upright standing

Purpose: To quantify the effect of unstable shoe wearing on muscle activity and haemodynamic response during standing. Methods: Thirty volunteers were divided into 2 groups: the experimental group wore an unstable shoe for 8 weeks, while the control group used a conventional shoe for the same period. Muscle activity of the medial gastrocnemius, tibialis anterior, rectus femoris and biceps femoris and venous circulation were assessed in quiet standing with the unstable shoe and barefoot. Results: In the first measurement there was an increase in medial gastrocnemius activity in all volunteers while wearing the unstable shoe. On the other hand, after wearing the unstable shoe for eight weeks these differences were not verified. Venous return increased in subjects wearing the unstable shoe before and after training. Conclusions: The unstable shoe produced changes in electromyographic characteristics which were advantageous for venous circulation even after training accommodation by the neuromuscular system.

Influence of environmental temperature and humidity on the acute ventilatory response to exercise in asthmatic adolescents

Asthma is a chronic inflammatory disorder of the respiratory airways affecting people of all ages, and constitutes a serious public health problem worldwide (6). Such a chronic inflammation is invariably associated with injury and repair of the bronchial epithelium known as remodelling (11). Inflammation, remodelling, and altered neural control of the airways are responsible for both recurrent exacerbations of asthma and increasingly permanent airflow obstruction (11, 29, 34). Excessive airway narrowing is caused by altered smooth muscle behaviour, in close interaction with swelling of the airway walls, parenchyma retractile forces, and enhanced intraluminal secretions (29, 38). All these functional and structural changes are associated with the characteristic symptoms of asthma – cough, chest tightness, and wheezing –and have a significant impact on patients’ daily lives, on their families and also on society (1, 24, 29). Recent epidemiological studies show an increase in the prevalence of asthma, mainly in industrial countries (12, 25, 37). The reasons for this increase may depend on host factors (e.g., genetic disposition) or on environmental factors like air pollution or contact with allergens (6, 22, 29). Physical exercise is probably the most common trigger for brief episodes of symptoms, and is assumed to induce airflow limitations in most asthmatic children and young adults (16, 24, 29, 33). Exercise-induced asthma (EIA) is defined as an intermittent narrowing of the airways, generally associated with respiratory symptoms (chest tightness, cough, wheezing and dyspnoea), occurring after 3 to 10 minutes of vigorous exercise with a maximal severity during 5 to 15 minutes after the end of the exercise (9, 14, 16, 24, 33). The definitive diagnosis of EIA is confirmed by the measurement of pre- and post-exercise expiratory flows documenting either a 15% fall in the forced expiratory volume in 1 second (FEV1), or a ≥15 to 20% fall in peak expiratory flow (PEF) (9, 24, 29). Some types of physical exercise have been associated with the occurrence of bronchial symptoms and asthma (5, 15, 17). For instance, demanding activities such as basketball or soccer could cause more severe attacks than less vigorous ones such as baseball or jogging (33). The mechanisms of exercise-induced airflow limitations seem to be related to changes in the respiratory mucosa induced by hyperventilation (9, 29). The heat loss from the airways during exercise, and possibly its post-exercise rewarming may contribute to the exercise-induced bronchoconstriction (EIB) (27). Additionally, the concomitant dehydration from the respiratory mucosa during exercise leads to an increased interstitial osmolarity, which may also contribute to bronchoconstriction (4, 36). So, the risk of EIB in asthmatically predisposed subjects seems to be higher with greater ventilation rates and the cooler and drier the inspired air is (23). The incidence of EIA in physically demanding coldweather sports like competitive figure skating and ice hockey has been found to occur in up to 30 to 35% of the participants (32). In contrast, swimming is often recommended to asthmatic individuals, because it improves the functionality of respiratory muscles and, moreover, it seems to have a concomitant beneficial effect on the prevalence of asthma exacerbations (14, 26), supporting the idea that the risk of EIB would be smaller in warm and humid environments. This topic, however, remains controversial since the chlorified water of swimming pools has been suspected as a potential trigger factor for some asthmatic patients (7, 8, 20, 21). In fact, the higher asthma incidence observed in industrialised countries has recently been linked to the exposition to chloride (7, 8, 30). Although clinical and epidemiological data suggest an influence of humidity and temperature of the inspired air on the bronchial response of asthmatic subjects during exercise, some of those studies did not accurately control the intensity of the exercise (2, 13), raising speculation of whether the experienced exercise overload was comparable for all subjects. Additionally, most of the studies did not include a control group (2, 10, 19, 39), which may lead to doubts about whether asthma per se has conditioned the observed results. Moreover, since the main targeted age group of these studies has been adults (10, 19, 39), any extrapolation to childhood/adolescence might be questionable regarding the different lung maturation. Considering the higher incidence of asthma in youngsters (30) and the fact that only the works of Amirav and coworkers (2, 3) have focused on this age group, a scarcity of scientific data can be identified. Additionally, since the main environmental trigger factors, i.e., temperature and humidity, were tested separately (10, 28, 39) it would be useful to analyse these two variables simultaneously because of their synergic effect on water and heat loss by the airways (31, 33). It also appears important to estimate the airway responsiveness to exercise within moderate environmental ranges of temperature and humidity, trying to avoid extreme temperatures and humidity conditions used by others (2, 3). So, the aim of this study was to analyse the influence of moderate changes in air temperature and humidity simultaneously on the acute ventilatory response to exercise in asthmatic children. To overcome the above referred to methodological limitations, we used a 15 minute progressive exercise trial on a cycle ergometer at 3 different workload intensities, and we collected data related to heart rate, respiratory quotient, minute ventilation and oxygen uptake in order to ensure that physiological exercise repercussions were the same in both environments. The tests were done in a “normal” climatic environment (in a gymnasium) and in a hot and humid environment (swimming pool); for the latter, direct chloride exposition was avoided.

Man-tailored biomimetic sensor of molecularly imprinted materials for the potentiometric measurement of oxytetracycline

A novel biomimetic sensor for the potentiometric transduction of oxytetracycline is presented. The artificial host was imprinted in methacrylic acid and/or acrylamide based polymers. Different amounts of molecularly imprinted and non-imprinted polymers were dispersed in different plasticizing solvents and entrapped in a poly(vinyl chloride) matrix. Only molecularly imprinted based sensors allowed a potentiometric transduction, suggesting the existence of host–guest interactions. These sensors exhibited a near-Nernstian response in steady state evaluations; slopes and detection limits ranged 42–63 mV/decade and 2.5–31.3 µg/mL, respectively. Sensors were independent from the pH of test solutions within 2–5. Good selectivity was observed towards glycine, ciprofloxacin, creatinine, acid nalidixic, sulfadiazine, cysteine, hydroxylamine and lactose. In flowing media, the biomimetic sensors presented good reproducibility (RSD of ±0.7%), fast response, good sensitivity (65 mV/decade), wide linear range (5.0×10−5 to 1.0×10−2 mol/L), low detection limit (19.8 µg/mL), and a stable baseline for a 5×10−3M citrate buffer (pH 2.5) carrier. The sensors were successfully applied to the analysis of drugs and urine. This work confirms the possibility of using molecularly imprinted polymers as ionophores for organic ion recognition in potentiometric transduction.

Differences in Mismatch Negativity (MMN) response to Pure-tone and Speech sounds in normal subjects: an additional explanation

The relation of automatic auditory discrimination, measured with MMN, with the type of stimuli has not been well established in the literature, despite its importance as an electrophysiological measure of central sound representation. In this study, MMN response was elicited by pure-tone and speech binaurally passive auditory oddball paradigm in a group of 8 normal young adult subjects at the same intensity level (75 dB SPL). The frequency difference in pure-tone oddball was 100 Hz (standard = 1 000 Hz; deviant = 1 100 Hz; same duration = 100 ms), in speech oddball (standard /ba/; deviant /pa/; same duration = 175 ms) the Portuguese phonemes are both plosive bi-labial in order to maintain a narrow frequency band. Differences were found across electrode location between speech and pure-tone stimuli. Larger MMN amplitude, duration and higher latency to speech were verified compared to pure-tone in Cz and Fz as well as significance differences in latency and amplitude between mastoids. Results suggest that speech may be processed differently than non-speech; also it may occur in a later stage due to overlapping processes since more neural resources are required to speech processing.

Forest soil short term response to prescribed fire: an approach by Principal Components Analysis

In the current context of serious climate changes, where the increase of the frequency of some extreme events occurrence can enhance the rate of periods prone to high intensity forest fires, the National Forest Authority often implements, in several Portuguese forest areas, a regular set of measures in order to control the amount of fuel mass availability (PNDFCI, 2008). In the present work we’ll present a preliminary analysis concerning the assessment of the consequences given by the implementation of prescribed fire measures to control the amount of fuel mass in soil recovery, in particular in terms of its water retention capacity, its organic matter content, pH and content of iron. This work is included in a larger study (Meira-Castro, 2009(a); Meira-Castro, 2009(b)). According to the established praxis on the data collection, embodied in multidimensional matrices of n columns (variables in analysis) by p lines (sampled areas at different depths), and also considering the quantitative data nature present in this study, we’ve chosen a methodological approach that considers the multivariate statistical analysis, in particular, the Principal Component Analysis (PCA ) (Góis, 2004). The experiments were carried out in a soil cover over a natural site of Andaluzitic schist, in Gramelas, Caminha, NW Portugal, who was able to maintain itself intact from prescribed burnings from four years and was submit to prescribed fire in March 2008. The soils samples were collected from five different plots at six different time periods. The methodological option that was adopted have allowed us to identify the most relevant relational structures inside the n variables, the p samples and in two sets at the same time (Garcia-Pereira, 1990). Consequently, and in addition to the traditional outputs produced from the PCA, we have analyzed the influence of both sampling depths and geomorphological environments in the behavior of all variables involved.