Binomial Theorem Research Articles

Computation for the Summation of Binomial Expansions and Geometric Series of Multiples of Powers of Two

Computation for the Summation of Binomial Expansions and Geometric Series of Multiples of Powers of Two

Novel Rational Approximated Fractional Order Lead Compensator

This paper proposes a simplified and analytical method for designing fractional order lead compensator (FOLC). In the proposed method, the structure of fractional order lead compensator considered is more generalized. The fractional operator of the FOLC is approximated to its equivalent rational form using binomial expansion to lower the computation complexity of the designing approach. Then, steady-state gain, gain cross over frequency and phase margin are used as the design specifications of the FOLC. The proposed method also considers the magnitude and phase of the plant to be controlled. Phase margin criteria ensure the stability of the designed compensator. The compensator is designed strictly adhering to gain and phase margin criteria. The proposed methodology is validated with four different illustrative examples through simulation. Integral absolute error (IAE) and integral square error (ISE) are considered as the performance indices. The simulation results demonstrate that the proposed scheme gives a better transient response, IAE, and ISE as compared to the existing FOLC design techniques.

Relation of Some Known Functions in terms of Generalized Meijer G -Functions

The aim of this paper is to prove some identities in the form of generalized Meijer G -function. We prove the relation of some known functions such as exponential functions, sine and cosine functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansion, logarithmic function, and sine integral, with the generalized Meijer G -function. We also prove the product of modified Bessel function of first and second kind in the form of generalized Meijer G -function and solve an integral involving the product of modified Bessel functions.

The SCAFFOLDING AND INVESTIGATION OF THINKING TO CALCULUS PROJECT IN PROJECT BASED LEARNING (PBL) ACTIVITIES

PBL adoption is increasingly widespread and recognized for various disciplines in higher education and in basic education has made a trend in learning innovation today. PBL provides individual freedom to find, utilize, build knowledge through experience in the form of project completion. The characteristics of the project used are the existence of investigations that give students the freedom to change and build knowledge. The investigation took place every day in a row from the student team (2 people) for 4 days for the given calculus project. The results of project completion investigations were delivered through team presentations, discussions, interviews with lecturers, and the provision of scaffolding by lecturers.Investigation which success include combination, definition of maclaurin series, tribe and series and derivative function which support project finishing, describing material supporting of project through example,verification binomial theorem, identification to 30 tribes maclaurin series/binomial series and approach value through tool of conting/ software(excel,calculator,geogebra) Scaffolding thinks incompatibility occurs in understanding and interpretation of combination notation, thought theorem, and implementation in calculations. Researcher's discovery that students are very active in building knowledge and exploring their experiences, as well as building openness in teams and instructors.

Mathematics Education Students’ Understanding of Binomial Series Expansion Based on the APOS Theory

Undergraduate students study the topic of binomial series expansion as part of their Calculus course. The purpose of this study was to explore the mental constructions of binomial series expansion of a class of 159 students. Data were collected through a written assessment task by each member of the class. A convenient sample of eleven students from the class was selected to undertake semi-structured interviews. The Action-Process-Object-Schema theory which was used depicts development of understanding of mathematics concepts through the hierarchical growth of mental constructions. Students in the study coped well with questions requiring action-level engagements but grappled with those calling for higher-level mental constructions, namely, processes and objects. Developing an object conception for a mathematical concept is the pinnacle for all teaching but also the most difficult to develop. Thus, this study underlines the significance of helping students construct object conceptions in binomial series expansion.

МЕТОДИКА ПРЕПОДАВАНИЯ КОНЕЧНОЙ МАТЕМАТИКИ В ПЕДАГОГИЧЕСКИХ УНИВЕРСИТЕТАХ В СВЯЗИ С НОВЫМИ ТЕНДЕНЦИЯМИ РАЗВИТИЯ МАТЕМАТИЧЕСКИХ НАУК

There were introduced new methods of the teaching and instruction of the following parts of the Pre-calculus: (1) Binomial Series; (2) Trigonometry; (3) Partial Fractions. The problems introduced in the article for the Pre-Calculus Course in Finite Mathematics was developed by the author. These unabridged problems are developed within the new trends in the evolutions of the novelty of the syllabi in Mathematics due to the development of the Mathematics Sciences / Theory and Applications. These new trends in the Theory and Application of Mathematics Sciences have been added new demands to the newly revised textbooks and corresponding syllabi for the Mathematics Courses taught at the Junior two years Colleges and Pedagogical Universities.These newly developed problems are reflection of the Development of Mathematical and Engineering Sciences to offer great amount of learning conclusion/sequel to those who pursue a bachelor’s degree at the universities of the pedagogical orientation. The problems presented in the article here are developed and restructured in terms of the newly developed techniques to solve the problem in Finite Mathematics and Engineering sciences. Moreover, the techniques offered in the article here are more likely to get utilized in Advanced Engineering Sciences, too, within the content of the problems, which require to obtain finite numerical solutions to the Real Phenomena Natural Problems in Engineering Sciences and Applied Problems in Mathematical Physics.The aim of this present publication is to offer new advanced techniques and instructional strategies to discuss methodology and instructional strategies of the mathematical training of the students at the Pedagogical Universities. Moreover, these new teaching techniques and strategies introduced may be extended to the engineering sciences at the technical universities, too.The results and scientific novelty of the introduced methodology and learning conclusions and sequel of the new knowledge the students at the Pedagogical universities may be benefited from are in the following list of the learning conclusions, presented in the article here. The students of the pedagogical orientation may attain the mastery skills in the following sections of the combinatorics in Finite Mathematics subject:- The n! Combination of n different terms.- Evaluate the expressions with factorials.- Identify that there are -!!( )!nrnr various of combinations of r identical terms in n variations.- Identify and evaluate the combinatorial coefficients from the Binomial Theorem.- Identify and able to build the Pascal’s triangle of the binomial coefficients.- Utilize the Binomial Theorem to expand the binomial formula for any natural powers.- Utilize the Binomial Theorem to obtain the general formula for the n-th term of binomial expansion.- Utilize the Sigma Symbols in the Binomial Theorem for the n-th terms of the binomial expansion. 19- Generate the expansion for the power of the ex, where e is the base of natural logarithmic function y = f(x) = x.Practical significance: the methods of teaching and new teaching strategies offered here in the article alongside with the application of the new trends in the development of mathematical and mathematics education sciences can be useful for prospective and currently practicing teachers of mathematics. Moreover, the materials presented here in this article can be useful for the educational professionals in their professional development plans to improve the quality in education

105.42 Generalised binomial theorem via Laplace transform technique

105.42 Generalised binomial theorem via Laplace transform technique

A Combinatorial Reliability Analysis of Generic Service Function Chains in Data Center Networks

In data center networks, the reliability of Service Function Chain (SFC)—an end-to-end service presented by a chain of virtual network functions (VNFs)—is a complex and specific function of placement, configuration, and application requirements, both in hardware and software. Existing approaches to reliability analysis do not jointly consider multiple features of system components, including, (i) heterogeneity, (ii) disjointness, (iii) sharing, (iv) redundancy, and (v) failure interdependency. To this end, we develop a novel analysis of service reliability of the so-called generic SFC, consisting of n = k + r sub-SFCs, whereby k ≥ 1 and r ≥ 0 are the numbers of arbitrary placed primary and backup (redundant) sub-SFCs, respectively. Our analysis is based on combinatorics and a reduced binomial theorem—resulting in a simple approach, which, however, can be utilized to analyze rather complex SFC configurations. The analysis is practically applicable to various VNF placement strategies in arbitrary data center configurations, and topologies and can be effectively used for evaluation and optimization of reliable SFC placements.

On the Fibonacci quaternion sequence with quadruple-produce components

This paper examines the Fibonacci quaternion sequence with quadruple-produce components, and demonstrates a golden-like ratio and some identities for this sequence. Its generating and exponential generating functions are given. Along with these, its series and binomial sum formula are established.

Non-linear ladder operators and coherent states for the 2:1 oscillator

The 2:1 two-dimensional anisotropic quantum harmonic oscillator is considered and new sets of states are defined by means of normal-ordering non-linear operators through the use of non-commutative binomial theorems as well as solving recurrence relations. The states generated are good candidates for the natural generalisation of the coherent states of the two-dimensional isotropic oscillator. The two-dimensional non-linear generalised ladder operators lead to several chains of states which are connected in a non trivial way. The uncertainty relations of the defining chain of states are calculated and it is found that they admit a resolution of the identity and the spatial distribution of the wavefunction produces Lissajous figures in correspondence with the classical 2:1 oscillator.

Blade coating analysis of a viscoelastic Carreau fluid using Adomian decomposition method

In this paper, a mathematical model of the blade coating of the viscoelastic Carreau fluid which passes through the gap between the fixed blade and the moving substrate is constructed. The governing equations in the light of LAT (Lubrication Approximation Theory) are nondimensionalized and the solutions for the velocity, volumetric flow rate and the pressure gradient are calculated using Adomian decomposition method and numerical method. Numerical results for the velocity distribution using shooting method have also been studied by taking the full lubrication equation without taking the binomial expansion. Some important quantities from the industrial point of view like pressure, shear stress, Nusselt number and skin friction coefficient are also calculated. The important results are graphically shown at the end. With exponential blade coater, we have calculated velocity profiles, pressure, coating thickness and blade loading. Some of these results are shown graphically while other are presented in the tabulated form.

Calculation of the Probability Distribution of CIBS Score in Different Relationships and Its Application.

Objective To derive the probability distribution formula of combined identity by state （CIBS） score among individuals with different relationships based on population data of autosomal multiallelic genetic markers. Methods The probabilities of different identity by state （IBS） scores occurring at a single locus between two individuals with different relationships were derived based on the principle of ITO method. Then the distribution probability formula of CIBS score between two individuals with different relationships when a certain number of genetic markers were used for relationship identification was derived based on the multinomial distribution theory. The formula was compared with the CIBS probability distribution formula based on binomial distribution theory. Results Between individuals with a certain relationship, labelled as RS, the probabilities of IBS=2, 1 and 0 occurring at a certain autosomal genetic marker x （that is, p2（RSx）, p1（RSx） and p0（RSx））, can be calculated based on the allele frequency data of that genetic marker and the probability of two individuals with the corresponding RS relationship sharing 0, 1 or 2 identity by descent （IBD） alleles （that is, φ0, φ1 and φ2）. For a genotyping system with multiple independent genetic markers, the distribution of CIBS score between pairs of individuals with relationships other than parent-child can be deducted using the averages of the 3 probabilities of all genetic markers （that is, p2（RS）, p1（RS） and p0（RS））, based on multinomial distribution theory. Conclusion The calculation of CIBS score distribution formula can be extended to all kinships and has great application value in case interpretation and system effectiveness evaluation. In most situations, the results based on binomial distribution formula are similar to those based on the formula derived in this study, thus, there is little difference between the two methods in actual work.

Decentralized Cooperative Online Estimation With Random Observation Matrices, Communication Graphs and Time Delays

We analyze convergence of decentralized cooperative online estimation algorithms by a network of multiple nodes via information exchanging in an uncertain environment. Each node has a linear observation of an unknown parameter with randomly time-varying observation matrices. The underlying communication network is modeled by a sequence of random digraphs and is subjected to nonuniform random time-varying delays in channels. Each node runs an online estimation algorithm consisting of a consensus term taking a weighted sum of its own estimate and neighbours' delayed estimates, and an innovation term processing its own new measurement at each time step. By stochastic time-varying system, martingale convergence theories and the binomial expansion of random matrix products, we transform the convergence analysis of the algorithm into that of the mathematical expectation of random matrix products. Firstly, for the delay-free case, we show that the algorithm gains can be designed properly such that all nodes' estimates converge to the true parameter in mean square and almost surely if the observation matrices and communication graphs satisfy the stochastic spatio-temporal persistence of excitation condition. Secondly, for the case with time delays, we introduce delay matrices to model the random time-varying communication delays between nodes. It is shown that under the stochastic spatio-temporal persistence of excitation condition, for any given bounded delays, proper algorithm gains can be designed to guarantee mean square convergence for the case with conditionally balanced digraphs.

A refinement of the binomial distribution using the quantum binomial theorem

q-analogs of special functions, including hypergeometric functions, play a central role in mathematics and have numerous applications in physics. In the theory of probability, q-analogs of various probability distributions have been introduced over the years, including the binomial distribution. Here, I propose a new refinement of the binomial distribution by way of the quantum binomial theorem (also known as the noncommutative q-binomial theorem), where the q is a formal variable in which information related to the sequence of successes and failures in the underlying binomial experiment is encoded in its exponent.

A detailed description of the binomial theorem and an application to permutation binomials over finite fields

In this paper, we present a detailed description of the binomial theorem and obtain some new classes of combinatorial identities. As an application, we discuss a class of permutation binomials over finite fields $${\mathbb {F}}_q$$ , which is of the form $$x^{\mu +\nu }+2x^{\mu }$$ , where $$q\equiv 1\pmod {3}$$ and $$(\nu , q-1)=\frac{q-1}{3}$$ .

An approach to the magnetic field of a finite solenoid with a circular cross-section

We present a simple approach, different from approaches in the literature, for calculating the magnetic field of a finite solenoid with a circular cross section, carrying an ideal surface current with an ordinary azimuthal component and a small axial component due to nonideal winding. The results are given as an infinite series whose terms are elementary functions or simple polynomials rather than other infinite series. It is easy to deal with the results because the polynomials involved are partial binomial expansions. These series converge rapidly everywhere, except near the edges of the solenoid. From these results, it is very easy to derive some general properties of the magnetic field, to reduce the field to greatly simplified forms in various special regions, and to address the special cases of infinite and semi-infinite solenoids. The azimuthal field due to nonideal winding is shown to be small everywhere. A solenoid with a small but finite thickness in the radial direction is investigated in detail, and we present simple analytic approximate results for its magnetic field. A comparison with an exact numerical evaluation shows that these are very good approximations.

Considering Instructional Approach & Question Design with the Hardy-Weinberg Principle

The Hardy-Weinberg principle (HWP) is an application of the binomial expansion theorem that is foundational to the field of population genetics. Because of the important history of the HWP in answering how variation is preserved during evolution, and the ability of Hardy-Weinberg equilibrium (HWE) to detect natural and sexual selection acting on a trait, the HWP is a staple of the introductory biology undergraduate curriculum in the United States. Introductory courses often cover a wide range of topics in ecology and evolution, and it is important that students have enough time during the semester to grasp the foundations of population genetics. At the same time, information needs to be presented clearly to ensure that the student gains a correct understanding of the HWP. This article discusses the importance of the HWP to undergraduate education in biology and describes misconceptions from the instructor’s perspective. These misconceptions are pervasive and risk undermining a proper understanding of the HWP. We provide examples adapted from university- and AP-level standardized tests.

Investment decisions under incomplete markets in the presence of wealth effects

This paper investigates the impact of market incompleteness on investment decisions. In contrast to previous studies, we assume an entrepreneur has power utility and investment causes an additive increase in her wealth. By using the binomial theorem, we derive the analytical solutions for the option value and investment threshold. First, we show that an increase in wealth alleviates the over-investment problem caused by the market incompleteness. In addition, the marginal value of wealth is always more than one and decreases in the wealth level. Finally, the increase in wealth substantially enhances the value of the investment option and lowers the idiosyncratic risk premium.

New set of non-separable 2D and 3D invariant moments for image representation and recognition

The extraction of feature vectors of the images is the most important step in image recognition, because it allows a good description of the forms to be recognized. This step is an operation that makes it possible to convert an image into a vector of real or complex values that can serve as a signature for this image. For an accurate recognition system, the used feature vector must be invariant to the three image transformations (translation, rotation and scale), which means that, the descriptor vectors of the image and the transformed image by translation, rotation or scale must be equal. Given the importance of moments and their invariants in pattern recognition and imaging, based on Newton’s binomial and trinomial formulas and the normalized central moments, we construct in this paper four new series of moments for 2D and 3D image recognition, which are invariant to translation, scaling and rotation (TSR): the first is a set of non-orthogonal moment invariants for 2D images (2dMIs), the second is a set of orthogonal moment invariants for 2D images (2dOMIs), the third is a set of non-orthogonal moment invariants for 3D images (3dMIs) and the fourth is a set of orthogonal moment invariants for 3D images (3dOMIs). Using the proposed invariant moments (2dMIs), (2dOMIs), (3dMIs) and (3dOMIs), we construct two types of image descriptor vectors for 2D images and two types for 3D images, which are invariant to translation, rotation and scale. A series of experiments is performed to validate this new set of invariant moments and compare its performance with the existing invariants moments. The obtained results ensure the superiority of the proposed moments over all existing moments in image recognition. Experiments on processing time show that the proposed method is faster than the existing orthogonal invariant moments.

Powers of Octonions

As it is known, Binomial expansion, De Moivre’s formula, and Euler’s formula are suitable methods for computing the powers of a complex number, but to compute the powers of an octonion number in easy way, we need to derive suitable formulas from these methods. In this paper, we present a novel way to compute the powers of an octonion number using formulas derived from the binomial expansion.