Vladimir Kulish | Nanyang Technological University (original) (raw)

Papers by Vladimir Kulish

Spontaneous transitions of genomic DNA segments from right-handed B-DNA into the left-handed, hig... more 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.

Cancer is a class of diseases characterized by out-of-control cells' growth which affect cells an... more Cancer is a class of diseases characterized by out-of-control cells' growth which affect cells and make them damaged. Many treatment options for cancer exist. Chemotherapy as an important treatment option is the use of drugs to treat cancer. The anticancer drug travels to the tumor and then diffuses in it through capillaries. The diffusion of drugs in the solid tumor is limited by penetration depth which is different in case of different drugs and cancers. The computation of this depth is important as it helps physicians to investigate about treatment of infected tissue. Although many efforts have been made on studying and measuring drug penetration depth, less works have been done on computing this length from a mathematical point of view. In this paper, first we propose phase lagging model for diffusion of drug in the tumor. Then, using this model on one side and considering the classic diffusion on the other side, we compute the drug penetration depth in the solid tumor. This computed value of drug penetration depth is corroborated by comparison with the values measured by experiments.

Cancer is a class of diseases characterized by out-of-control cells' growth which affect DNAs and... more 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. Cells production and die are regulated in human body in an orderly way. But in case of cancer, the division and growth of cells is out of control. In this manner, the damaged cells start to occupy more and more space in a part of body and so they expel the useful healthy cells. By this way that part of body is called tumor. So fighting with these cancer cells and changing the way of their production and accumulation is a critical issue in medical science. Scientists have developed different methods for treatment of cancer. Some of these methods are surgery , chemotherapy, radiation therapy, hormonal therapy, targeted therapy and palliative care. Employing the less-invasive methods have always had critical role in patient treatment. Chemotherapy as a method for cancer treatment deals with application of drugs affecting the cancer cell's ability to divide and reproduce. The drug makes the cancer cells weak and destroys them by directly applying to cancer site or through the bloodstream. During years some researchers have worked on mathematical modelling of the effect of chemotherapy on cancer treatment. Some researchers employed different types of differential equations for modelling of the effect of chemotherapy on cancer treatment. For instance, Pillis et al. developed a mathematical model based on a system of ordinary differential equations which analyses the cancer growth on a cell population level after using chemotherapy 1. Despite the overall success of this mathematical model, it couldn't explain the effects of IL-2 on a growing tumour. So, in another work Pillis et al. updated their model by introducing new parameters governing their values from empirical data which are specific in case of different patients. The new model allows production of endogenous IL-2, IL-2-stimulated NK cell proliferation and IL-2-dependent CD8 + T-cell self-regulations 2. In another work, using a system of delayed differential equations, Liu and Freedman proposed a mathematical model of vascular tumor treatment using chemotherapy. This model represents the number of blood vessels within the tumor and changes in mass of healthy cells and competing parenchyma cells. Using the proposed model they considered a continuous treatment for tumor growth 3. See also 4,5. In a closer approach some researchers specially focused on mathematical modelling of the diffusion of anti-cancer drugs to cancer tumor 6–11 .

This paper derives a new integral relationship between heat flux and temperature in a transient, ... more 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.

A novel macroscopic gas transport model, derived from fundamental engineering principles , is use... more A novel macroscopic gas transport model, derived from fundamental engineering principles , is used to simulate the three-dimensional, unsteady respiration process within the alveolar region of the lungs. The simulations, mimicking the single-breath technique for measuring the lung diffusing capacity for carbon-monoxide (CO), allow the prediction of the red blood cell (RBC) distribution effects on the lung diffusing capacity. Results, obtained through numerical simulations, unveil a strong relationship between the type of distribution and the lung diffusing capacity. Several RBC distributions are considered, namely: normal (random), uniform, center-cluster, and corner-cluster red cell distributions. A nondimensional correlation is obtained in terms of a geometric parameter characterizing the RBC distribution, and presented as a useful tool for predicting the RBC distribution effect on the lung diffusing capacity. The effect of red cell movement is not considered in the present study because CO does not equilibrate with capillary blood within the time spent by blood in the capillary. Hence, blood flow effect on CO diffusion is expected to be only marginal.

—Among the approaches used to analyze biomedical signals, e.g. electroencephalogram, there are me... more —Among the approaches used to analyze biomedical signals, e.g. electroencephalogram, there are methods based on fractal dimension (FD) calculation. One of the main drawbacks of these methods is the requirement for long duration of the analyzed signal. To analyze events of brief duration in real-time mode and to apply the results obtained directly in the time domain, thus providing an easier interpretation of FD behavior; in this work we present a fast algorithm to calculate the FD of the signal. It is derived from the generalized entropy, namely Renyi entropy.

In this paper we detail a phase lagging model of brain response to external stimuli. The model is... more In this paper we detail a phase lagging model of brain response to external stimuli. The model is derived using the basic laws of physics like conservation of energy law. This model eliminates the paradox of instantaneous propagation of the action potential in the brain. The solution of this model is then presented. The model is further applied in the case of a single neuron and is verified by simulating a single action potential. The results of this modeling are useful not only for the fundamental understanding of single action potential generation, but also they can be applied in case of neuronal interactions, where the results can be verified against the real EEG signal.

The paper presents generalized relation between the local values of temperature and the correspon... more The paper presents generalized relation between the local values of temperature and the corresponding heat flux in a one-dimensional 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.

This paper discusses the 50-Hz artifact observed during EEG recording of three patients, using Mi... more This paper discusses the 50-Hz artifact observed during EEG recording of three patients, using Mindset24 equipment in a biomedical laboratory along with other instruments. The artifact at 50 Hz is created due to electromagnetic (EM) radiation from the environment. Perhaps, the nullification of 50 Hz artifact occurs by the three important means such as (a) grounding of far-field EM current when the patient touches the equipment metal casing, (b) connecting the electrodes in a sequence to obtain differential potentials with suppressed near-field EM radiation potentials of the scalp, and (c) destructive interferences of the EM waves.

Human brain response is the overall ability of the brain in analyzing internal and external stimu... more Human brain response is the overall ability of the brain in analyzing internal and external stimuli in the form of transferred energy to the mind/brain phase-space and thus, making the proper decisions. During the last decade scientists discovered 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 was less effort which have been done on the mathematical modeling of the human brain response to external stimuli. This research is devoted to the modeling of human EEG signal, as an alert state of overall human brain activity monitoring, due to receiving external stimuli, based on fractional diffusion equation. The results of this modeling show very good agreement with the real human EEG signal and thus, this model can be used as a strong representative of the human brain activity.

A novel mathematical model of micro socio-economic behavior is proposed. It is based on the new c... more A novel mathematical model of micro socio-economic behavior is proposed. It is based on the new concept of the informational channel, viewed as an elementary socio-economic cell. The processes, taking place within this cell, are described by a set of partial differential equations that relate the two main aspects of a socio-economic system: energy (any material asset, such as money, material property, etc.) and information. The solution is obtained in the form of the Volterra integral equation that relates the value of disturbance upon the socio-economical cell and the value of the wealth function, which is defined as an integral characteristic of both material and informational components. Two basic cases of disturbance were analyzed: a constant and a single-pulse flux. In both cases, the results show a quite natural behavior. The proposed model might serve as a basis for more advanced analysis of socio-economical systems.

From the text: Waves follow our boat as we meander across the lake, and turbulent air currents fo... more From the text: Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations. Thus, the papers published in this special issue address topics related to the Navier-Stokes equations, transition to turbulence, and some other related problems. These papers cover topics in applied mathematics, and their applications to science and engineering.

The method of Laplace transform is standard in mathematical physics. One of its advantages is tha... more The method of Laplace transform is standard in mathematical physics. One of its advantages is that it is still applicable in the case of PDE with fractional derivatives. But the situation becomes much more complicated if the rate of differentiation can vary with time, even if these variations have very small amplitude. In this paper, a theorem is proved which gives the expression for the Laplace transform of the diffusion equation with time-dependent rate of time derivative. Such equations appear in biology and financial mathematics. It follows that in case of variable fractional derivatives the method of Laplace transform can be used only under certain conditions as approximate method.

Causality and Locality in Modern Physics, 1998

The existence of the velocity potential is a direct consequence from the derivation of the contin... more The existence of the velocity potential is a direct consequence from the derivation of the continuity equation from the Schroedinger equation. This implies that the Cole-Hopf transformation is applicable to the Navier-Stokes equation for an incompressible flow and allows reducing the Navier-Stokes equation to the Einstein-Kolmogorov equation, in which the reaction term depends on the pressure. The solution to the resulting equation, and to the Navier-Stokes equation as well, can then be written in terms of the Green's function of the heat equation and is given in the form of an integral mapping. Such a form of the solution makes bifurcation period doubling possible, i.e. solutions to transition and turbulent flow regimes in spite of the existence of the velocity potential.

Scientific reports, 2015

Cancer is a class of diseases characterized by out-of-control cells' growth which affect DNAs... more 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.

Consciousness is variously defined and discussed in literature from different points of view such... more Consciousness is variously defined and discussed in literature from different points of view such as biology, neuropsychology, etc. During last decade scientists discovered the human consciousness and proposed some models defining this phenomenon. Despite rapid advances in studies related to this area of brain research, there is no general mathematical equation which defines human consciousness in the whole brain scale. This research is devoted to study of the human consciousness as the result of human perception in order to develop some mathematical equations which are defined in macroscopic level of brain organization. The results of this research are useful not only for the fundamental understanding of human consciousness, but shall be very useful in the areas of brain research such as seizure prediction using brain signals.

Spontaneous transitions of genomic DNA segments from right-handed B-DNA into the left-handed, hig... more 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.

Cancer is a class of diseases characterized by out-of-control cells' growth which affect cells an... more Cancer is a class of diseases characterized by out-of-control cells' growth which affect cells and make them damaged. Many treatment options for cancer exist. Chemotherapy as an important treatment option is the use of drugs to treat cancer. The anticancer drug travels to the tumor and then diffuses in it through capillaries. The diffusion of drugs in the solid tumor is limited by penetration depth which is different in case of different drugs and cancers. The computation of this depth is important as it helps physicians to investigate about treatment of infected tissue. Although many efforts have been made on studying and measuring drug penetration depth, less works have been done on computing this length from a mathematical point of view. In this paper, first we propose phase lagging model for diffusion of drug in the tumor. Then, using this model on one side and considering the classic diffusion on the other side, we compute the drug penetration depth in the solid tumor. This computed value of drug penetration depth is corroborated by comparison with the values measured by experiments.

Cancer is a class of diseases characterized by out-of-control cells' growth which affect DNAs and... more 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. Cells production and die are regulated in human body in an orderly way. But in case of cancer, the division and growth of cells is out of control. In this manner, the damaged cells start to occupy more and more space in a part of body and so they expel the useful healthy cells. By this way that part of body is called tumor. So fighting with these cancer cells and changing the way of their production and accumulation is a critical issue in medical science. Scientists have developed different methods for treatment of cancer. Some of these methods are surgery , chemotherapy, radiation therapy, hormonal therapy, targeted therapy and palliative care. Employing the less-invasive methods have always had critical role in patient treatment. Chemotherapy as a method for cancer treatment deals with application of drugs affecting the cancer cell's ability to divide and reproduce. The drug makes the cancer cells weak and destroys them by directly applying to cancer site or through the bloodstream. During years some researchers have worked on mathematical modelling of the effect of chemotherapy on cancer treatment. Some researchers employed different types of differential equations for modelling of the effect of chemotherapy on cancer treatment. For instance, Pillis et al. developed a mathematical model based on a system of ordinary differential equations which analyses the cancer growth on a cell population level after using chemotherapy 1. Despite the overall success of this mathematical model, it couldn't explain the effects of IL-2 on a growing tumour. So, in another work Pillis et al. updated their model by introducing new parameters governing their values from empirical data which are specific in case of different patients. The new model allows production of endogenous IL-2, IL-2-stimulated NK cell proliferation and IL-2-dependent CD8 + T-cell self-regulations 2. In another work, using a system of delayed differential equations, Liu and Freedman proposed a mathematical model of vascular tumor treatment using chemotherapy. This model represents the number of blood vessels within the tumor and changes in mass of healthy cells and competing parenchyma cells. Using the proposed model they considered a continuous treatment for tumor growth 3. See also 4,5. In a closer approach some researchers specially focused on mathematical modelling of the diffusion of anti-cancer drugs to cancer tumor 6–11 .

This paper derives a new integral relationship between heat flux and temperature in a transient, ... more 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.

A novel macroscopic gas transport model, derived from fundamental engineering principles , is use... more A novel macroscopic gas transport model, derived from fundamental engineering principles , is used to simulate the three-dimensional, unsteady respiration process within the alveolar region of the lungs. The simulations, mimicking the single-breath technique for measuring the lung diffusing capacity for carbon-monoxide (CO), allow the prediction of the red blood cell (RBC) distribution effects on the lung diffusing capacity. Results, obtained through numerical simulations, unveil a strong relationship between the type of distribution and the lung diffusing capacity. Several RBC distributions are considered, namely: normal (random), uniform, center-cluster, and corner-cluster red cell distributions. A nondimensional correlation is obtained in terms of a geometric parameter characterizing the RBC distribution, and presented as a useful tool for predicting the RBC distribution effect on the lung diffusing capacity. The effect of red cell movement is not considered in the present study because CO does not equilibrate with capillary blood within the time spent by blood in the capillary. Hence, blood flow effect on CO diffusion is expected to be only marginal.

—Among the approaches used to analyze biomedical signals, e.g. electroencephalogram, there are me... more —Among the approaches used to analyze biomedical signals, e.g. electroencephalogram, there are methods based on fractal dimension (FD) calculation. One of the main drawbacks of these methods is the requirement for long duration of the analyzed signal. To analyze events of brief duration in real-time mode and to apply the results obtained directly in the time domain, thus providing an easier interpretation of FD behavior; in this work we present a fast algorithm to calculate the FD of the signal. It is derived from the generalized entropy, namely Renyi entropy.

In this paper we detail a phase lagging model of brain response to external stimuli. The model is... more In this paper we detail a phase lagging model of brain response to external stimuli. The model is derived using the basic laws of physics like conservation of energy law. This model eliminates the paradox of instantaneous propagation of the action potential in the brain. The solution of this model is then presented. The model is further applied in the case of a single neuron and is verified by simulating a single action potential. The results of this modeling are useful not only for the fundamental understanding of single action potential generation, but also they can be applied in case of neuronal interactions, where the results can be verified against the real EEG signal.

The paper presents generalized relation between the local values of temperature and the correspon... more The paper presents generalized relation between the local values of temperature and the corresponding heat flux in a one-dimensional 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.

This paper discusses the 50-Hz artifact observed during EEG recording of three patients, using Mi... more This paper discusses the 50-Hz artifact observed during EEG recording of three patients, using Mindset24 equipment in a biomedical laboratory along with other instruments. The artifact at 50 Hz is created due to electromagnetic (EM) radiation from the environment. Perhaps, the nullification of 50 Hz artifact occurs by the three important means such as (a) grounding of far-field EM current when the patient touches the equipment metal casing, (b) connecting the electrodes in a sequence to obtain differential potentials with suppressed near-field EM radiation potentials of the scalp, and (c) destructive interferences of the EM waves.

Human brain response is the overall ability of the brain in analyzing internal and external stimu... more Human brain response is the overall ability of the brain in analyzing internal and external stimuli in the form of transferred energy to the mind/brain phase-space and thus, making the proper decisions. During the last decade scientists discovered 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 was less effort which have been done on the mathematical modeling of the human brain response to external stimuli. This research is devoted to the modeling of human EEG signal, as an alert state of overall human brain activity monitoring, due to receiving external stimuli, based on fractional diffusion equation. The results of this modeling show very good agreement with the real human EEG signal and thus, this model can be used as a strong representative of the human brain activity.

A novel mathematical model of micro socio-economic behavior is proposed. It is based on the new c... more A novel mathematical model of micro socio-economic behavior is proposed. It is based on the new concept of the informational channel, viewed as an elementary socio-economic cell. The processes, taking place within this cell, are described by a set of partial differential equations that relate the two main aspects of a socio-economic system: energy (any material asset, such as money, material property, etc.) and information. The solution is obtained in the form of the Volterra integral equation that relates the value of disturbance upon the socio-economical cell and the value of the wealth function, which is defined as an integral characteristic of both material and informational components. Two basic cases of disturbance were analyzed: a constant and a single-pulse flux. In both cases, the results show a quite natural behavior. The proposed model might serve as a basis for more advanced analysis of socio-economical systems.

From the text: Waves follow our boat as we meander across the lake, and turbulent air currents fo... more From the text: Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations. Thus, the papers published in this special issue address topics related to the Navier-Stokes equations, transition to turbulence, and some other related problems. These papers cover topics in applied mathematics, and their applications to science and engineering.

The method of Laplace transform is standard in mathematical physics. One of its advantages is tha... more The method of Laplace transform is standard in mathematical physics. One of its advantages is that it is still applicable in the case of PDE with fractional derivatives. But the situation becomes much more complicated if the rate of differentiation can vary with time, even if these variations have very small amplitude. In this paper, a theorem is proved which gives the expression for the Laplace transform of the diffusion equation with time-dependent rate of time derivative. Such equations appear in biology and financial mathematics. It follows that in case of variable fractional derivatives the method of Laplace transform can be used only under certain conditions as approximate method.

Causality and Locality in Modern Physics, 1998

The existence of the velocity potential is a direct consequence from the derivation of the contin... more The existence of the velocity potential is a direct consequence from the derivation of the continuity equation from the Schroedinger equation. This implies that the Cole-Hopf transformation is applicable to the Navier-Stokes equation for an incompressible flow and allows reducing the Navier-Stokes equation to the Einstein-Kolmogorov equation, in which the reaction term depends on the pressure. The solution to the resulting equation, and to the Navier-Stokes equation as well, can then be written in terms of the Green's function of the heat equation and is given in the form of an integral mapping. Such a form of the solution makes bifurcation period doubling possible, i.e. solutions to transition and turbulent flow regimes in spite of the existence of the velocity potential.

Scientific reports, 2015

Cancer is a class of diseases characterized by out-of-control cells' growth which affect DNAs... more 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.

Consciousness is variously defined and discussed in literature from different points of view such... more Consciousness is variously defined and discussed in literature from different points of view such as biology, neuropsychology, etc. During last decade scientists discovered the human consciousness and proposed some models defining this phenomenon. Despite rapid advances in studies related to this area of brain research, there is no general mathematical equation which defines human consciousness in the whole brain scale. This research is devoted to study of the human consciousness as the result of human perception in order to develop some mathematical equations which are defined in macroscopic level of brain organization. The results of this research are useful not only for the fundamental understanding of human consciousness, but shall be very useful in the areas of brain research such as seizure prediction using brain signals.