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Since the brain controls heart activations, there should be a correlation between their activities in different conditions. This study investigates the correlation between heart and brain responses to olfactory stimulation. We employed fractal theory and sample entropy to evaluate the complexity of EEG signals and Heart Rate Variability (HRV) in the form of R–R time series. We applied four different pleasant odors with different molecular complexities to 13 participants and analyzed their EEG and ECG signals. The results demonstrated that the complexities of HRV and EEG signals are strongly correlated; a bigger alteration in the complexity of olfactory stimuli is mapped to a bigger alteration in the complexity of HRV and EEG signals. This investigation can be similarly done to examine the correlation between various organs and the brain by quantifying the complexity of their signals versus brain signals.

BACKGROUND Education and learning are the most important goals of all universities. For this purpose, lecturers use various tools to grab the attention of students and improve their learning ability. Virtual reality refers to the subjective sensory experience of being immersed in a computer-mediated world, and has recently been implemented in learning environments. OBJECTIVE The aim of this study was to analyze the effect of a virtual reality condition on students’ learning ability and physiological state. METHODS Students were shown 6 sets of videos (3 videos in a two-dimensional condition and 3 videos in a three-dimensional condition), and their learning ability was analyzed based on a subsequent questionnaire. In addition, we analyzed the reaction of the brain and facial muscles of the students during both the two-dimensional and three-dimensional viewing conditions and used fractal theory to investigate their attention to the videos. RESULTS The learning ability of students was increased in the three-dimensional condition compared to that in the two-dimensional condition. In addition, analysis of physiological signals showed that students paid more attention to the three-dimensional videos. CONCLUSIONS A virtual reality condition has a greater effect on enhancing the learning ability of students. The analytical approach of this study can be further extended to evaluate other physiological signals of subjects in a virtual reality condition.

Bulk metallic glasses (BMGs) are disordered compounds without a long-range crystalline order. In this paper, we study the effect of an element’s standard entropy on the BMG’s entropy and complexity for four elements. In order to do this, we simulate the BMG structures first. Then we compute the entropy and fractal dimension of the resulting structures as measures of disorder and complexity, respectively. We verify our predictions via experiments. In doing so, we substitute the elements in separate experiments and compute the fractal dimension of the obtained BMG pattern. The results of our investigations show that there is a relation between the variations of elements’ entropies and the variations of BMGs’ entropy and fractality. On the other hand, measurement of the compressive strength for the obtained BMGs from experiments shows that by increasing the entropy and fractal dimension, BMG’s compressive strength increases. The adaptive capability observed in this method could potentially be harnessed for targeted BMG production, depending on the specific design goal. The method discussed here is not only useful for analysis of BMG structures, but it can be applied to other structures.

One of the major challenges in brain research is to relate the structural features of the auditory stimulus to structural features of Electroencephalogram (EEG) signal. Memory content is an important feature of EEG signal and accordingly the brain. On the other hand, the memory content can also be considered in case of stimulus. Beside all works done on analysis of the effect of stimuli on human EEG and brain memory, no work discussed about the stimulus memory and also the relationship that may exist between the memory content of stimulus and the memory content of EEG signal. For this purpose we consider the Hurst exponent as the measure of memory. This study reveals the plasticity of human EEG signals in relation to the auditory stimuli. For the first time we demonstrated that the memory content of an EEG signal shifts towards the memory content of the auditory stimulus used. The results of this analysis showed that an auditory stimulus with higher memory content causes a larger increment in the memory content of an EEG signal. For the verification of this result, we benefit from approximate entropy as indicator of time series randomness. The capability, observed in this research, can be further investigated in relation to human memory.

One of the major challenges in vision research is to analyze the effect of visual stimuli on human vision. However, no relationship has been yet discovered between the structure of the visual stimulus, and the structure of fixational eye movements. This study reveals the plasticity of human fixational eye movements in relation to the ‘complex’ visual stimulus. We demonstrated that the fractal temporal structure of visual dynamics shifts towards the fractal dynamics of the visual stimulus (image). The results showed that images with higher complexity (higher fractality) cause fixational eye movements with lower fractality. Considering the brain, as the main part of nervous system that is engaged in eye movements, we analyzed the governed Electroencephalogram (EEG) signal during fixation. We have found out that there is a coupling between fractality of image, EEG and fixational eye movements. The capability observed in this research can be further investigated and applied for treatment of different vision disorders.

One of the major challenges in olfaction research is to relate the structural features of the odorants to different features of olfactory system. However, no relationship has been yet discovered between the structure of the olfactory stimulus, and the structure of respiratory signal. This study reveals the plasticity of human respiratory signal in relation to 'complex' olfactory stimulus (odorant). We demonstrated that fractal temporal structure of respiration dynamics shifts towards the properties of the odorants used. The results show for the first time that more structurally complex a monomolecular odorant will result in less fractal respiratory signal. On the other hand, odorant with higher entropy will result the respiratory signal with lower entropy. The capability observed in this research can be further investigated and applied for treatment of patients with different respiratory diseases.

An important challenge in heart research is to make the relation between the features of external stimuli and heart activity. Olfactory stimulation is an important type of stimulation that affects the heart activity, which is mapped on Electrocardiogram (ECG) signal. Yet, no one has discovered any relation between the structures of olfactory stimuli and the ECG signal. This study investigates the relation between the structures of heart rate and the olfactory stimulus (odorant). We show that the complexity of the heart rate is coupled with the molecular complexity of the odorant, where more structurally complex odorant causes less fractal heart rate. Also, odorant having higher entropy causes the heart rate having lower approximate entropy. The method discussed here can be applied and investigated in case of patients with heart diseases as the rehabilitation purpose.

Cancer is a class of diseases characterized by out-of-control cells’ growth which affect DNAs and make them damaged. Many treatment options for cancer exist, with the primary ones including surgery, chemotherapy, radiation therapy, hormonal therapy, targeted therapy and palliative care. Which treatments are used depends on the type, location, and grade of the cancer as well as the person’s health and wishes. Chemotherapy is the use of medication (chemicals) to treat disease. More specifically, chemotherapy typically refers to the destruction of cancer cells. Considering the diffusion of drugs in cancer cells and fractality of DNA walks, in this research we worked on modelling and prediction of the effect of chemotherapy on cancer cells using Fractional Diffusion Equation (FDE). The employed methodology is useful not only for analysis of the effect of special drug and cancer considered in this research but can be expanded in case of different drugs and cancers.

Skin cancer is a common, low-grade cancerous (malignant) growth of the skin. It starts from cells that begin as normal skin cells and transform into those with the potential to reproduce in an out-of-control manner. Cancer develops when DNA, the molecule found in cells that encodes genetic information, becomes damaged and the body cannot repair the damage. A DNA walk of a genome represents how the frequency of each nucleotide of a pairing nucleotide couple changes locally. In this research in order to diagnose the skin cancer, first DNA walk plots of genomes of patients with skin cancer were generated. Then, the data so obtained was checked for complexity by computing the fractal dimension. Furthermore, the Hurst exponent has been employed in order to study the correlation of damaged DNA. By analysing different samples it has been found that the damaged DNA sequences are exhibiting higher degree of complexity and less correlation compared to normal DNA sequences. This investigation confirms that this method can be used for diagnosis of skin cancer. The method discussed in this research is useful not only for diagnosis of skin cancer but can be applied for diagnosis and growth analysis of different types of cancers. MOE (Min. of Education, S’pore) Published version

In previous work, the exact solution to the Navier-Stokes equation has been established. The solution is written in the integral form and provides a relationship between the local values of the velocity and corresponding shear stress. The solution is in the form of an integral mapping, which should allow chaotic solutions (transition to turbulence) under some set of parameters. This possibility is investigated in the present paper.

In previous work, the generalised relation between the local values of temperature and the corresponding heat flux has been achieved by the use of a novel technique that involves generalised derivatives (in particular, derivatives of non-integer orders). Confluent hyper-geometric functions, known as Whittaker’s functions, appear in the course of the solution procedure, upon applying the Laplace transform to the original transport equation. The relation is written in the integral form and provides a relationship between the local values of the temperature and heat flux. In the case of a non-zero value of the boundary speed, the solution is in the form of an integral mapping, which should allow chaotic solutions under some set of parameters. This possibility is investigated in the present paper.

The paper is focused on the development of a thermodynamic mathematical model describing the internal ballistic phenomena, taking place within working chambers of an airgun. The objective is to derive a governing equation modeling the change of gas temperature in a control volume subsuming several incoming and outgoing mass airflow rates. A novel algorithm for solving internal ballistics of all types of airguns is also developed and applied for the case of spring-piston airgun. The mathematical model and solving algorithm are validated and verified experimentally.

In this paper, a new phenomenological model of heat transfer in hard-disk air bearing based on the nonlocal behavior in air molecular gas is developed. The model predicts that the heat transfer process between a slider and disk surface occurs due to heat conduction and the viscous dissipation for Couette and Poiseuille flows. The assumption of the nonlocal behavior (in space) in air molecular gas, due to its scattering from the boundaries of the domain through which air flow is moved, allows us to eliminate deficiencies of the continuum model, in particular, the one based on the temperature jump theory. As a result, the proposed model is more accurate for simulating heat transfer in hard-disk air bearing.

Carbon dioxide is widely used as the power gas in the gas guns community due to its ease of handling, storability at room temperature, and high vapor pressure depending only upon temperature, but not a tank size, as long as some liquid carbon dioxide remains in the tank. This high vapor pressure can be used as the pressurant, making it what is referred to as a self-pressurising propellant. However, as a two-phase substance, carbon dioxide does have its drawbacks: (1) vaporization of liquefied CO2 inside a tank when shooting rapidly or a lot causes the tank to get cool, resulting in pressure fluctuations that makes the gun’s performance and accuracy worse, (2) solid carbon dioxide that is also known as dry ice can appear on the output valve of the tank while shooting and it can cause damage or slow the gun’s performance down, if it works its way into some control components, including the barrel of the gun. Hence, it is crucial to obtain a scientific understanding of carbon dioxide behavior and further the discharge characteristics of a wide range of pressure-tank configurations. For the purpose of satisfying this goal, a comprehensive discharge mathematical model for carbon dioxide tank dynamics is required. In this paper, the possibility to develop an advanced non-equilibrium model of depressurization in two-phase fluids is discussed. MOE (Min. of Education, S’pore) Published version

It was Zel’dovich, who first considered the transient combustion problem of solid propellants. Some more detailed models of that process have been developed afterwards. However, until today, numerical methods remain the prevailing tool for modeling unsteady combustion processes. In this work, it has been demonstrated that at least one of the problems of the unsteady combustion theory, which previously investigated numerically, can be treated analytically by means of fractional calculus. The solution for the unsteady speed of combustion thus derived is then compared with the solution obtained by numerical means in previous studies. The comparison shows a good agreement between those results, especially for small values of time.It was Zel’dovich, who first considered the transient combustion problem of solid propellants. Some more detailed models of that process have been developed afterwards. However, until today, numerical methods remain the prevailing tool for modeling unsteady combustion processes. In this work, it has been demonstrated that at least one of the problems of the unsteady combustion theory, which previously investigated numerically, can be treated analytically by means of fractional calculus. The solution for the unsteady speed of combustion thus derived is then compared with the solution obtained by numerical means in previous studies. The comparison shows a good agreement between those results, especially for small values of time.

An important challenge in heart research is to make the relation between the features of external stimuli and heart activity. Olfactory stimulation is an important type of stimulation that affects the heart activity, which is mapped on Electrocardiogram (ECG) signal. Yet, no one has discovered any relation between the structures of olfactory stimuli and the ECG signal. This study investigates the relation between the structures of heart rate and the olfactory stimulus (odorant). We show that the complexity of the heart rate is coupled with the molecular complexity of the odorant, where more structurally complex odorant causes less fractal heart rate. Also, odorant having higher entropy causes the heart rate having lower approximate entropy. The method discussed here can be applied and investigated in case of patients with heart diseases as the rehabilitation purpose.

One of the major challenges in olfaction research is to relate the structural features of the odorants to different features of olfactory system. However, no relationship has been yet discovered between the structure of the olfactory stimulus and the structure of respiratory signal. This study reveals the plasticity of human respiratory signal in relation to ‘complex’ olfactory stimulus (odorant). We demonstrated that fractal temporal structure of respiration dynamics shifts towards the properties of the odorants used. The results show for the first time that more structurally complex a monomolecular odorant will result in less fractal respiratory signal. On the other hand, odorant with higher entropy will result the respiratory signal with lower entropy. The capability observed in this research can be further investigated and applied for treatment of patients with different respiratory diseases.

One of the major challenges in vision research is to analyze the effect of visual stimuli on human vision. However, no relationship has been yet discovered between the structure of the visual stimulus and the structure of fixational eye movements. This study reveals the plasticity of human fixational eye movements in relation to the ‘complex’ visual stimulus. We demonstrated that the fractal temporal structure of visual dynamics shifts towards the fractal dynamics of the visual stimulus (image). The results showed that images with higher complexity (higher fractality) cause fixational eye movements with lower fractality. Considering the brain, as the main part of nervous system that is engaged in eye movements, we analyzed the governed Electroencephalogram (EEG) signal during fixation. We have found out that there is a coupling between fractality of image, EEG and fixational eye movements. The capability observed in this research can be further investigated and applied for treatment of different vision disorders.

The paper presents generalized relation between the local values of temperature and the corresponding heat flux in a onedimensional semi-infinite domain with the moving boundary. The generalized relation between the local values of temperature and the corresponding heat flux has been achieved by the use of a novel technique that involves generalized derivatives (in particular, derivatives of non-integer orders). Confluent hyper-geometric functions, known as Whittaker’s functions, appear in the course of the solution procedure, upon applying the Laplace transform to the original transport equation. The relation is written in the integral form and provides a relationship between the local values of the temperature and heat flux. Accepted version

Heat transfer processes, induced by ultrashort laser pulses in thin gold films, were studied with a time resolution of 50 fs. It is demonstrated that in thin gold films heat is transmitted by means of electron–phonon and phonon–phonon interactions, and dissipated on nanoscale within 800 fs. Measurements show that the electron–phonon relaxation time varies versus the probe wavelength from 1.6 to 0.8 ps for λ = 560–630 nm. Applied mathematical model is a result of transforming the two-temperature model to the hyperbolic heat equation, based on assumptions that the electron gas is heated up instantaneously and applying Cattaneo’s law to the phonon subsystem, agrees well with the experimental results. This model allows us to define time of electron–phonon scattering as the ratio of the heat penetration depth to the speed of sound in the bulk material that, in turn, provides an explanation of experimental results that show the dependence of the electron–phonon relaxation time on the wavelength. Accepted version

Analysis of human movements is an important category of research in biomedical engineering, especially for the rehabilitation purpose. The movement of limbs is investigated usually by analyzing the movement signals. Less efforts have been made to investigate how neural that correlate to the movements, are represented in the human brain. In this research, for the first time we decode the limb movements by fractal analysis of Electroencephalogram (EEG) signals. We investigated how the complexity of EEG signal changes in different limb movements in motor execution (ME), and motor imagination (MI) sessions. The result of our analysis showed that the EEG signal experiences greatest level of complexity in elbow flexion and hand-close movements in ME, and MI sessions respectively. On the other hand, the lowest level of complexity of EEG signal belongs to hand-open and rest condition in ME, and MI sessions, respectively. Employing fractal theory in analysis of bio signals is not limited to EEG signal, and can be further investigated in other types of human’s bio signals in different conditions. The result of these investigations can vastly been employed for the rehabilitation purpose.

Post-match wheelchair rugby performance analysis is currently confined to single parameters (e.g. distance, average speed, activity time) without presenting the change of these parameters with time in the course of a match. The aim of this study is to determine a single activity parameter, which represents the instantaneous performance and which serves for comparing the performance between different players, as well as to develop a method which is relatively cheap and ubiquitously applicable. A rugby wheelchair was instrumented with an accelerometer (iPhone by Apple) and data were collected while pushing at all of the four combinations of high/low acceleration and high/low frequency. These four combinations were assigned performance ranks from 1 to 4, and the following activity parameters were determined: mean acceleration amplitude, mean stroke frequency, and the Hausdorff (fractal) dimension. The Hausdorff dimension correlated highly with the mean acceleration amplitude, mean frequency and performance rank as well as with the product of frequency and amplitude. The Hausdorff dimension of the signal is suitable to replace and represent conventional performance parameters.

This paper derives a new integral relationship between heat flux and temperature in a transient, one-dimensional heat conducting half-space having an arbitrary initial condition. A unified mathematical treatment is proposed that is extendable to higher-dimensional and finite-region geometries. This relationship is developed by combining Green's functions method with a formalism associated with the regularization of weakly singular, integral equations. The resulting analytic expression provides the in-depth, heat flux history through the time integration of a weakly singular integral involving the heating/cooling rate, dT/dt at a single spatial position. If temperature data are collected at the embedded location then resulting formulation is mildly ill posed and hence requires digital filtering for removing high-frequency noise from the collected signal. On the other hand, if heating/cooling rate data are collected at the embedded location then the resulting expression is well posed. From this formalism, a new heat flux sensor strategy is produced based on the temporal variable and not the spatial variable as used in Gardon gauges. In particular, Gardon heat flux gauges are based on Fourier's law involving a spatial gradient.

The paper presents a novel mathematical model of price evolution within the market. The model accounts for a finite time lag between the disturbance (news) and the market response to it. It follows from the model that any single disturbance brings the market out of its efficient state and the market is to come back into this state after some finite relaxation time. A quantitative measure of the market efficiency–the market efficiency coefficient–has been introduced. It has been demonstrated that a phase-lagging market is never either fully efficient or fully inefficient due to the finiteness of the frequency of disturbances (news) acting upon the market.

Spontaneous transitions of genomic DNA segments from right-handed B-DNA into the left-handed, high-energy Z conformation are unstable within topologically relaxed DNA molecules, such as mammalian chromosomes. Here we show, from direct application of the principles of statistical physics with a promoter region in the mouse genome as a representative example, that the life span for this alternate DNA conformation may be much smaller than the characteristic time of thermal fluctuations that cause the B-to-Z transition. Surprisingly, such a short existence of Z-DNA is important because it can be responsible for super-transport of energy within a genome. This type of energy transport can be utilized by a cell to communicate information about the state of particular chromatin domains within chromosomes or as a buffer against genome instability.

Human gait is defined as human locomotion that is achieved through the movement of limbs. Different limb movement patterns result in different gait patterns. Different internal and external stimuli can affect the human gait. During the years, scientists have worked on the analysis of the effect of external stimuli on human gait, but no work has been reported yet that suggests a mathematical model for analysis of this effect by linking to the nervous system. Considering the diffusion of external stimuli to the nervous system on one side and fractality of human gait on another side, in this research, for the first time, we develop a model for prediction of human gait using fractional diffusion equation. Using this model, we will predict the effect of auditory stimuli on human gait. The model developed in this research is useful not only for analysis of the effect of auditory stimuli on human gait but can be used also for analysis of different types of stimuli on subjects with healthy conditions or having some types of diseases. Copyright © 2016 John WileySons, Ltd.

Cancer is a class of diseases characterized by out-of-control cells’ growth which affect DNAs and make them damaged. Many treatment options for cancer exist, with the primary ones including surgery, chemotherapy, radiation therapy, hormonal therapy, targeted therapy and palliative care. Which treatments are used depends on the type, location and grade of the cancer as well as the person’s health and wishes. Chemotherapy is the use of medication (chemicals) to treat disease. More specifically, chemotherapy typically refers to the destruction of cancer cells. Considering the diffusion of drugs in cancer cells and fractality of DNA walks, in this research we worked on modelling and prediction of the effect of chemotherapy on cancer cells using Fractional Diffusion Equation (FDE). The employed methodology is useful not only for analysis of the effect of special drug and cancer considered in this research but can be expanded in case of different drugs and cancers.

The paper presents a novel technique of nonlinear spectral analysis. This technique is based on the concept of generalized entropy of a given probability distribution, known as the Rényi entropy. This concept allows defining generalized fractal dimension of encephalogram (EEG) and determining fractal spectra of encephalographic signals. These spectra contain information of both frequency and amplitude characteristics of EEG and can be used together with well-accepted techniques of EEG analysis as an enhancement of the latter. Powered by volume visualization of the brain activity, the method provides new clues for understanding the mental processes in humans.

An integral relation is obtained between local temperature and local temperature gradient for the dual-lag model of heat transport, which substitutes classical Fourier law at short time scales. Both the heat flux lag and the temperature gradient lag are considered, however, it is shown that only difference between the two affects temperature profile. Being applied at exposed surface of material, the integral equation predicts surface temperature variation for any form of imposed heat flux. The solution is tested considering solid heating by a picosecond laser impulse. Results are compared with the classical solution of the parabolic heat transfer equation and available experimental data.

The relationship between the local temperature and the local heat flux has been established for the homogeneous heat equation, which takes into account the dual-phase-lag. This relationship has been written in the form of a convolution integral, involving the modified Bessel functions. Thus, the integral equation obtained has been solved numerically for processes of surface heating with time scale in the order of picoseconds. The results have been compared with the results obtained from the classical equation of heat conduction (parabolic heat transfer), the hyperbolic heat transfer equation with single-phase-lag, and the numerical solution of the heat equation with dual-phase-lag, which were obtained using the finite difference method.

This paper presents an integral solution of the generalized one-dimensional equation of energy transport with the convective term.The solution of the problem has been achieved by the use of a novel technique that involves generalized derivatives (in particular, derivatives of noninteger orders). Confluent hypergeometric functions, known as Whittaker's functions, appear in the course of the solution procedure upon applying the Laplace transform to the original transport equation.The analytical solution of the problem is written in the integral form and provides a relationship between the local values of the transported property (e.g., temperature, mass, momentum, etc.) and its flux.The solution is valid everywhere within the domain, including the domain boundary.

The relationship between the local temperature and the local heat flux has been established for the homogeneous hyperbolic heat equation. This relationship has been written in the form of a convolution integral involving the modified Bessel functions. The scale analysis of the hyperbolic energy equation has been performed and the dimensionless criterion for the mode of energy transport, similar to the Reynolds criterion for the flow regimes, has been proposed. Finally, the integral equation, relating the local temperature and the local heat flux, has been solved numerically for those processes of surface heating whose time scale is of the order of picoseconds.

Human brain response is the result of the overall ability of the brain in analyzing different internal and external stimuli and thus making the proper decisions. During the last decades scientists have discovered more about this phenomenon and proposed some models based on computational, biological, or neuropsychological methods. Despite some advances in studies related to this area of the brain research, there were fewer efforts which have been done on the mathematical modeling of the human brain response to external stimuli. This research is devoted to the modeling and prediction of the human EEG signal, as an alert state of overall human brain activity monitoring, upon receiving external stimuli, based on fractional diffusion equations. The results of this modeling show very good agreement with the real human EEG signal and thus this model can be used for many types of applications such as prediction of seizure onset in patient with epilepsy. Published version

It has been shown previously that the strong solution to the Navier-Stokes equation for an incompressible fluid can be written in the form of an integral mapping that involves the Green’s function of the heat equation. In this work, it is demonstrated that the Reynolds number is the main parameter that controls the solution stability. MOE (Min. of Education, S’pore) Published version

In this note we present the application of fractional calculus, or the calculus of arbitrary (noninteger) differentiation, to the solution of time-dependent, viscous-diffusion fluid mechanics problems. Together with the Laplace transform method, the application of fractional calculus to the classical transient viscous-diffusion equation in a semi-infinite space is shown to yield explicit analytical (fractional) solutions for the shear-stress and fluid speed anywhere in the domain. Comparing the fractional results for boundary shear-stress and fluid speed to the existing analytical results for the first and second Stokes problems, the fractional methodology is validated and shown to be much simpler and more powerful than existing techniques.

The transient thermoreflectance (TTR) method consists of measuring changes in the reflectivity of a material (thin film) under pulsed laser heating, and relating these changes to the corresponding surface temperature variations. Analytical solutions of the diffusion problem are then used to determine the thermal conductivity of the material following an iterative matching process between the solutions and the experimental results. Analytical solutions are attainable either when the material absorbs the laser energy volumetrically or when the material absorbs the laser energy at the surface. Either solution allows for the determination of only one thermal property (thermal conductivity or diffusivity), with the other one assumed to be known. A new, single, analytical solution to the transient diffusion equation with simultaneous surface and volumetric heating, found using fractional calculus, is presented in a semi-derivative form. This complete solution provides the means to determine the two thermal properties of the material (thermal conductivity and diffusivity) concomitantly. In this preliminary study, the solution component for surface heating is validated by comparison with experimental data for a gold sample using the classical thermoreflectance method. Further results, for surface and volumetric heating, are obtained and analyzed considering a GaAs sample.