And I say to myself: “What a fractional world!”

This paper discusses several complex systems in the perspective of fractional dynamics. For prototype systems are considered the cases of deoxyribonucleic acid decoding, financial evolution, earthquakes events, global warming trend, and musical rhythms. The application of the Fourier transform and of the power law trendlines leads to an assertive representation of the dynamics and to a simple comparison of their characteristics. Moreover, the gallery of different systems, both natural and man made, demonstrates the richness of phenomena that can be described and studied with the tools of fractional calculus.

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.

Identifying economic periods and crisis with the multidimensional scaling

This paper applied MDS and Fourier transform to analyze different periods of the business cycle. With such purpose, four important stock market indexes (Dow Jones, Nasdaq, NYSE, S&P500) were studied over time. The analysis under the lens of the Fourier transform showed that the indexes have characteristics similar to those of fractional noise. By the other side, the analysis under the MDS lens identified patterns in the stock markets specific to each economic expansion period. Although the identification of patterns characteristic to each expansion period is interesting to practitioners (even if only in a posteriori fashion), further research should explore the meaning of such regularities and target to find a method to estimate future crisis.

Dynamics of the Dow Jones and the NASDAQ stock indexes

The goal of this study is the analysis of the dynamical properties of financial data series from worldwide stock market indices. We analyze the Dow Jones Industrial Average ( ∧ DJI) and the NASDAQ Composite ( ∧ IXIC) indexes at a daily time horizon. The methods and algorithms that have been explored for description of physical phenomena become an effective background, and even inspiration, for very productive methods used in the analysis of economical data. We start by applying the classical concepts of signal analysis, Fourier transform, and methods of fractional calculus. In a second phase we adopt a pseudo phase plane approach.

Complex dynamics in the trajectory control of redundant manipulators

Redundant manipulators allow the trajectory optimization, the obstacle avoidance, and the resolution of singularities. For this type of manipulators, the kinematic control algorithms adopt generalized inverse matrices that may lead to unpredictable responses. Motivated by these problems this paper studies the complexity revealed by the trajectory planning scheme when controlling redundant manipulators. The results reveal fundamental properties of the chaotic phenomena and give a deeper insight towards the development of superior trajectory control algorithms.

Fractional dynamics in the trajectory control of redundant manipulators

Under the pseudoinverse control, robots with kinematical redundancy exhibit an undesirable chaotic joint motion which leads to an erratic behavior. This paper studies the complexity of fractional dynamics of the chaotic response. Fourier and wavelet analysis provides a deeper insight, helpful to know better the lack of repeatability problem of redundant manipulators. This perspective for the study of the chaotic phenomena will permit the development of superior trajectory control algorithms.

On similarities in infrared spectra of complex drugs

A chromatographic separation of active ingredients of Combivir, Epivir, Kaletra, Norvir, Prezista, Retrovir, Trivizir, Valcyte, and Viramune is performed on thin layer chromatography. The spectra of these nine drugs were recorded using the Fourier transform infrared spectroscopy. This information is then analyzed by means of the cosine correlation. The comparison of the infrared spectra in the perspective of the adopted similarity measure is possible to visualize with present day computer tools, and the emerging clusters provide additional information about the similarities of the investigated set of complex drugs.

Rhapsody in fractional

This paper studies several topics related with the concept of “fractional” that are not directly related with Fractional Calculus, but can help the reader in pursuit new research directions. We introduce the concept of non-integer positional number systems, fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities, fractional delay discrete-time linear systems, and fractional Fourier transform.

Fractional calculus analysis of the cosmic microwave background

Cosmic microwave background (CMB) radiation is the imprint from an early stage of the Universe and investigation of its properties is crucial for understanding the fundamental laws governing the structure and evolution of the Universe. Measurements of the CMB anisotropies are decisive to cosmology, since any cosmological model must explain it. The brightness, strongest at the microwave frequencies, is almost uniform in all directions, but tiny variations reveal a spatial pattern of small anisotropies. Active research is being developed seeking better interpretations of the phenomenon. This paper analyses the recent data in the perspective of fractional calculus. By taking advantage of the inherent memory of fractional operators some hidden properties are captured and described.

Desenvolvimento de Anticorpos Plásticos para um Biomarcador do Cancro

O trabalho descrito compreende o desenvolvimento de um anticorpo plástico (MIP, do inglês Molecularly Imprinted Polymer) para o antigénio carcinoembrionário (CEA, do inglês Carcinoembriogenic Antigen) e a sua aplicação na construção de dispositivos portáteis, de tamanho reduzido e de baixo custo, tendo em vista a monitorização deste biomarcador do cancro do colo-retal em contexto Point-of-Care (POC). O anticorpo plástico foi obtido por tecnologia de impressão molecular orientada, baseada em eletropolimerização sobre uma superfície condutora de vidro recoberto por FTO. De uma forma geral, o processo foi iniciado pela electropolimerização de anilina sobre o vidro, seguindo-se a ligação por adsorção do biomarcador (CEA) ao filme de polianilina, com ou sem monómeros carregados positivamente (Cloreto de vinilbenziltrimetilamónio, VB). A última fase consistiu na electropolimerização de o-fenilenodiamina (oPD) sobre a superfície, seguindo-se a remoção da proteína por clivagem de ligações peptídicas, com o auxílio de tripsina. A eficiência da impressão do biomarcador CEA no material polimérico foi controlada pela preparação de um material análogo, NIP (do inglês, Non-Imprinted Polymer), no qual nem a proteína nem o monómero VB estavam presentes. Os materiais obtidos foram caracterizados quimicamente por técnicas de Infravermelho com Transformada de Fourier (FTIR, do inglês, Fourier Transform Infrared Spectroscopy) e microscopia confocal de Raman. Os materiais sensores preparados foram entretanto incluídos em membranas poliméricas de Poli(cloreto de vinilo) (PVC) plastificado, para construção de sensores (biomiméticos) seletivos a CEA, tendo-se avaliado a resposta analítica em diferentes meios. Obteve-se uma boa resposta potenciométrica em solução tampão de Ácido 4-(2-hidroxietil)piperazina-1-etanosulfónico (HEPES), a pH 4,4, com uma membrana seletiva baseada em MIP preparada com o monómero carregado VB. O limite de deteção foi menor do que 42 pg/mL, observando-se um comportamento linear (versus o logaritmo da concentração) até 625 pg/mL, com um declive aniónico igual a -61,9 mV/década e r2>0,9974. O comportamento analítico dos sensores biomiméticos foi ainda avaliado em urina, tendo em vista a sua aplicação na análise de CEA em urina. Neste caso, o limite de deteção foi menor do que 38 pg/mL, para uma resposta linear até 625 pg/mL, com um declive de -38,4 mV/década e r2> 0,991. De uma forma geral, a aplicação experimental dos sensores biomiméticos evidenciou respostas exatas, sugerindo que os biossensores desenvolvidos prossigam estudos adicionais tendo em vista a sua aplicação em amostras de indivíduos doentes.

Label-free human chorionic gonadotropin detection at picogram levels using oriented antibodies bound to graphene screen-printed electrodes

Human chorionic gonadotropin (hCG) is a key diagnostic marker of pregnancy and an important biomarker for cancers in the prostate, ovaries and bladder and therefore of great importance in diagnosis. For this purpose, a new immunosensor of screen-printed electrodes (SPEs) is presented here. The device was fabricated by introducing a polyaniline (PANI) conductive layer, via in situ electropolymerization of aniline, onto a screen-printed graphene support. The PANI-coated graphene acts as the working electrode of a three terminal electrochemical sensor. The working electrode is functionalised with anti-hCG, by means of a simple process that enabled oriented antibody binding to the PANI layer. The antibody was attached to PANI following activation of the –COOH group at the Fc terminal. Functionalisation of the electrode was analysed and optimized using Electrochemical Impedance Spectroscopy (EIS). Chemical modification of the surface was characterised using Fourier transform infrared, and Raman spectroscopy with confocal microscopy. The graphene–SPE–PANI devices displayed linear responses to hCG in EIS assays from 0.001 to 50 ng mL−1 in real urine, with a detection limit of 0.286 pg mL−1. High selectivity was observed with respect to the presence of the constituent components of urine (urea, creatinine, magnesium chloride, calcium chloride, sodium dihydrogen phosphate, ammonium chloride, potassium sulphate and sodium chloride) at their normal levels, with a negligible sensor response to these chemicals. Successful detection of hCG was also achieved in spiked samples of real urine from a pregnant woman. The immunosensor developed is a promising tool for point-of-care detection of hCG, due to its excellent detection capability, simplicity of fabrication, low-cost, high sensitivity and selectivity.

Electrochemical biosensor based on biomimetic material for myoglobin detection

A novel reusable molecularly imprinted polymer (MIP) assembled on a polymeric layer of carboxylated poly(vinyl chloride) (PVCsingle bondCOOH) for myoglobin (Myo) detection was developed. This polymer was casted on the gold working area of a screen printed electrode (Au-SPE), creating a novel disposable device relying on plastic antibodies. Electrochemical impedance spectroscopy (EIS), cyclic voltammetry (CV) and Fourier transform infrared spectroscopy (FTIR) studies confirmed the surface modification. The MIP/Au-SPE devices displayed a linear behaviour in EIS from 0.852 to 4.26 μg mL−1, of positive slope 6.50 ± 1.48 (kΩ mL μg−1). The limit of detection was 2.25 μg mL−1. Square wave voltammetric (SWV) assays were made in parallel and showed linear responses between 1.1 and 2.98 μg mL−1. A current decrease was observed against Myo concentration, producing average slopes of −0.28 ± 0.038 μA mL μg−1. MIP/Au-SPE also showed good results in terms of selectivity. The error% found for each interfering species were 7% for troponin T (TnT), 11% for bovine serum albumin (BSA) and 2% for creatine kinase MB (CKMB), respectively. Overall, the technical modification over the Au-SPE was found a suitable approach for screening Myo in biological fluids.

Rhapsody in Fractional

This paper studies several topics related with the concept of “fractional” that are not directly related with Fractional Calculus, but can help the reader in pursuit new research directions. We introduce the concept of non-integer positional number systems, fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities, fractional delay discrete-time linear systems, and fractional Fourier transform.

The potential of molecular imprinting as a biosensing devices for monitoring the CEA cancer biomarker

6th Graduate Student Symposium on Molecular Imprinting

The Persistence of Memory

This paper analyzes several natural and man-made complex phenomena in the perspective of dynamical systems. Such phenomena are often characterized by the absence of a characteristic length-scale, long range correlations and persistent memory, which are features also associated to fractional order systems. For each system, the output, interpreted as a manifestation of the system dynamics, is analyzed by means of the Fourier transform. The amplitude spectrum is approximated by a power law function and the parameters are interpreted as an underlying signature of the system dynamics. The complex systems under analysis are then compared in a global perspective in order to unveil and visualize hidden relationships among them.