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Sk. Sarif Hassan

Publications and source records attributed to Sk. Sarif Hassan.

17 recordsLinked to original sources

Fractality in a delayed-prey-predator ecological system with Allee effects

A delayed, discrete-time, prey-predator model with Allee effects imposed on prey and predator populations is defined, and dynamics of the system is characterized computationally. The parametric conditions for local asymptotic stability of equilibria are investigated with several illustrating examples. In addition, several periodic solutions were achieved. Furthermore, high degree of fractality of the prey-predator population trajectories is observed when the system is sufficiently delayed. This system has turned out to be a cooperative system, which is depicted using fractal dimension.

q-bio.PE↗

Relationship of Two Discrete Dynamical Models: One-dimensional Cellular Automata and Integral Value Transformations

Cellular Automaton (CA) and an Integral Value Transformation (IVT) are two well established mathematical models which evolve in discrete time steps. Theoretically, studies on CA suggest that CA is capable of producing a great variety of evolution patterns. However computation of non-linear CA or higher dimensional CA maybe complex, whereas IVTs can be manipulated easily. The main purpose of this paper is to study the link between a transition function of a one-dimensional CA and IVTs. Mathematically, we have also established the algebraic structures of a set of transition functions of a one-dimensional CA as well as that of a set of IVTs using binary operations. Also DNA sequence evolution has been modelled using IVTs.

nlin.CG↗

Two Dimensional Discrete Dynamics of Integral Value Transformations

A notion of dimension preservative map, \textit{Integral Value Transformations} (IVTs) is defined over $\mathbb{N}^k$ using the set of $p$-adic functions. Thereafter, two dimensional \textit{Integral Value Transformations} (IVTs) is systematically analyzed over $\mathbb{N} \times \mathbb{N}$ using pair of two variable Boolean functions. The dynamics of IVTs over $\mathbb{N} \times \mathbb{N}=\mathbb{N}^2$ is studied from algebraic perspective. It is seen that the dynamics of the IVTs solely depends on the dynamics (state transition diagram) of the pair of two variable Boolean functions. A set of sixteen \textit{Collatz-like} IVTs are identified in two dimensions. Also, the dynamical system of IVTs having attractor with one, two, three and four cycles are studied. Additionally, some quantitative information of \textit{Integral Value Transformations} (IVTs) in different bases and dimensions are also discussed.

math.DS↗

Dynamics of the complex rational delay recursive sequence $\displaystyle{z_{n+1}=\frac{α+βz_{n-k}}{γ- z_{n}}}$

Dynamics of the delay rational difference equation $\displaystyle{z_{n+1}=\frac{α+βz_{n-k}}{γ- z_{n}}}$ with complex parameters $α$, $β$, $γ$ and arbitrary complex initial conditions is investigated. Existence of prime period two solutions and higher order periods are ensured in the complex parameters unlike in the case of real parameters of the same rational difference equation. In addition, a new dynamical behavior, chaotic solutions of the difference equation are ensured computationally.

math.DS↗

Computational Complex Dynamics of $\displaystyle{f_{α, β, γ, δ}(z)=\frac{αz + β}{γz^2 +δz}}$

The dynamics of the family of maps $\displaystyle{f_{α, β, γ, δ}(z)=\frac{αz + β}{γz^2 +δz}}$ in complex plane is investigated computationally. This dynamical system $z_{n+1}=f_{α, β, γ, δ}(z_n)=\frac{αz_n + β}{γz_n^2 +δz_n}$ has periodic solutions with higher periods which was absent in the real line scenario. It is also found that there are chaotic fractal and non-fractal like solutions of the dynamical systems. A few special cases of parameters are also have been taken care.

math.DS↗

Understanding Functional Protein-Protein Interactions Of ABCB11 And ADA In Human And Mouse

Proteins are macromolecules which hardly act alone; they need to make interactions with some other proteins to do so. Numerous factors are there which can regulate the interactions between proteins [4]. Here in this present study we aim to understand Protein -Protein Interactions (PPIs) of two proteins ABCB11 and ADA from quantitative point of view. One of our major aims also is to study the factors that regulate the PPIs and thus to distinguish these PPIs with proper quantification across the two species Homo Sapiens and Mus Musculus respectively to know how one protein interacts with different set of proteins in different species.

q-bio.OT↗

Dynamics of $\displaystyle{z_{n+1}=\frac{α+ αz_{n}+βz_{n-1}}{1+z_{n}}}$ in Complex Plane

The dynamics of the second order rational difference equation $\displaystyle{z_{n+1}=\frac{α+ αz_{n}+βz_{n-1}}{1+z_{n}}}$ with complex parameters $α$, $β$ and arbitrary complex initial conditions is investigated. The same difference equation is well studied with positive real parameters and initial values and one of the main results on trichotomy of parameters is revisited in the complex set-up and the similar result is found to be true. In addition, the chaotic solutions of the equation is fetched out in the complex set-up which was absent in the real scenario.

math.DS↗

Complex Dynamics of $\displaystyle{z_{n+1}=\frac{α+ βz_{n}+ γz_{n-1}}{A + B z_n + C z_{n-1}}}$

The dynamics of the second order rational difference equation $\displaystyle{z_{n+1}=\frac{α+ βz_{n}+ γz_{n-1}}{A + B z_n + C z_{n-1}}}$ with complex parameters and arbitrary complex initial conditions is investigated. In the complex set up, the local asymptotic stability and boundedness are studied vividly for this difference equation. Several interesting characteristics of the solutions of this equation, using computations, which does not arise when we consider the same equation with positive real parameters and initial conditions are shown. The chaotic solutions of the difference equation is absolutely new feature in the complex set up which is also shown in this article. Some of the interesting observations led us to pose some open interesting problems regarding chaotic and higher order periodic solutions and global asymptotic convergence of this equation.

math.DS↗

Complex Dynamics of the Difference Equation $z_{n+1}=\fracα{z_{n}}+ \fracβ{z_{n-1}}$

The dynamics of the second order rational difference equation in the title with complex parameters and arbitrary complex initial conditions is investigated. Two associated difference equations are also studied. The solutions in the complex plane of such equations exhibit many rich and complicated asymptotic behavior. The analysis of the local stability of these three difference equations and periodicity have been carried out. We further exhibit several interesting characteristics of the solutions of this equation, using computations, which does not arise when we consider the same equation with positive real parameters and initial conditions. Many interesting observations led us to pose several open problems and conjectures of paramount importance regarding chaotic and higher order periodic solutions and global asymptotic convergence of such difference equations. It is our hope that these observations of these complex difference equations would certainly be new add-ons to the present art of research in rational difference equations in understanding the behavior in the complex domain.

math.DS↗

A Quantitative Understanding of Human Sex Chromosomal Genes

In the last few decades, the human allosomes are engrossed in an intensive attention among researchers. The allosomes are now already been sequenced and found there are about 2000 and 78 genes in human X and Y chromosomes respectively. The hemizygosity of the human X chromosome in males exposes recessive disease alleles, and this phenomenon has prompted decades of intensive study of X-linked disorders. By contrast, the small size of the human Y chromosome, and its prominent long-arm heterochromatic region suggested absence of function beyond sex determination. But the present problem is to accomplish whether a given sequence of nucleotides i.e. a DNA is a Human X or Y chromosomal genes or not, without any biological experimental support. In our perspective, a proper quantitative understanding of these genes is required to justify or nullify whether a given sequence is a Human X or Y chromosomal gene. In this paper, some of the X and Y chromosomal genes have been quantified in genomic and proteomic level through Fractal Geometric and Mathematical Morphometric analysis. Using the proposed quantitative model, one can easily make probable justification or deterministic nullification whether a given sequence of nucleotides is a probable Human X or Y chromosomal gene or not, without seeking any biological experiment. Of course, a further biological experiment is essential to validate it as the probable Human X or Y chromosomal gene homologue. This study would enable Biologists to understand these genes in more quantitative manner instead of their qualitative features.

q-bio.GN↗

Fractal and Mathematical Morphology in Intricate Comparison between Tertiary Protein Structures

Intricate comparison between two given tertiary structures of proteins is as important as the comparison of their functions. Several algorithms have been devised to compute the similarity and dissimilarity among protein structures. But, these algorithms compare protein structures by structural alignment of the protein backbones which are usually unable to determine precise differences. In this paper, an attempt has been made to compute the similarities and dissimilarities among 3D protein structures using the fundamental mathematical morphology operations and fractal geometry which can resolve the problem of real differences. In doing so, two techniques are being used here in determining the superficial structural (global similarity) and local similarity in atomic level of the protein molecules. This intricate structural difference would provide insight to Biologists to understand the protein structures and their functions more precisely.

cs.CG↗

A Spatial Crypto Technique for Secure Data Transmission

This paper presents a spatial encryption technique for secured transmission of data in networks. The algorithm is designed to break the ciphered data packets into multiple data which are to be packaged into a spatial template. A secure and efficient mechanism is provided to convey the information that is necessary for obtaining the original data at the receiver-end from its parts in the packets. An authentication code (MAC) is also used to ensure authenticity of every packet.

cs.CR↗

Use of L-system mathematics for making new subfamily members of olfactory receptor full length genes, OR1D2, OR1D4 and OR1D5

Ligands for only two human olfactory receptors are known. One of them, OR1D2, binds to Bourgeonal [Malnic B, Godfrey P-A, Buck L-B (2004) The human olfactory receptor gene family. Proc. Natl. Acad. Sci U. S. A. 101: 2584-2589 and Erratum in: Proc Natl Acad Sci U. S. A. (2004) 101: 7205]. OR1D2, OR1D4 and OR1D5 are three full length olfactory receptors present in an olfactory locus in human genome. These receptors are more than 80% identical in DNA sequences and have 108 base pair mismatches among them. We have used L-system mathematics and have been able to show a closely related subfamily of OR1D2, OR1D4 and OR1D5.

cs.OH↗

On the Interesting World of Fractals and Their Applications to Music

In this paper we have defined one function that has been used to construct different fractals having fractal dimensions between 1.58 and 2. Also, we tried to calculate the amount of increment of fractal dimension in accordance with the base of the number systems. Further, interestingly enough, these very fractals could be a frame of lyrics for the musicians, as we know that the fractal dimension of music is around 1.65 and varies between a high of 1.68 and a low of 1.60. Further, at the end we conjecture that the switching from one music fractal to another is nothing but enhancing a constant amount fractal dimension which might be equivalent to a kind of different sets of musical notes in various orientations.

cs.OH↗

On Universal Physical Reality in the Light of Quantum Consciousness

In this paper, we have first given an intuitive definition of "Consciousness" as realized by us. Next, from this intuitive definition we derived the physical definition of quantum consciousness (Quantum Consciousness Parameter or QCP). This QCP is the elementary level of consciousness in quantum particles, which are the most elementary particles in nature. Thus QCP can explain both the perceptible and non-perceptible nature and some existing postulates of physics. We conceptualize that the level of human consciousness is most complex having highest fractal dimension of 4.85 in the electroencephalographs experiment done by other research groups. On the other hand, other species are having lesser consciousness level, which can be reflected by lesser fractal dimensions. We have also explored the bio informatics of consciousness from genome viewpoints where we tried to draw an analogy of neurons with electrons and photons. Lastly, we refine the quantum mechanics in terms of QCP; we all know that in Einstein's special theory of relativity, Einstein has used the postulate "Consistency of the velocity of light irrespective of all frames of reference (inertial or non-inertial frames)". In our theoretical revelation QCP can be directly applied to get a confirmatory proof of this postulate. Thus the postulate can be framed as a law.

physics.gen-ph↗

Theory of Rule 6 and its Application to Round Robin Tournament

In this paper we have used one 2 variable Boolean function called Rule 6 to define another beautiful transformation named as Extended Rule-6. Using this function we have explored the algebraic beauties and its application to an efficient Round Robin Tournament (RRT) routine for 2k (k is any natural number) number of teams. At the end, we have thrown some light towards any number of teams of the form nk where n, k are natural numbers.

cs.DM↗

Theory of Carry Value Transformation (CVT) and its Application in Fractal formation

In this paper the theory of Carry Value Transformation (CVT) is designed and developed on a pair of n-bit strings and is used to produce many interesting patterns. One of them is found to be a self-similar fractal whose dimension is same as the dimension of the Sierpinski triangle. Different construction procedures like L-system, Cellular Automata rule, Tilling for this fractal are obtained which signifies that like other tools CVT can also be used for the formation of self-similar fractals. It is shown that CVT can be used for the production of periodic as well as chaotic patterns. Also, the analytical and algebraic properties of CVT are discussed. The definition of CVT in two-dimension is slightly modified and its mathematical properties are highlighted. Finally, the extension of CVT and modified CVT (MCVT) are done in higher dimensions.

cs.DM↗