Binomial Theorem Research Articles

On the longest/shortest negative excursion of a Lévy risk process and related quantities

In this paper, we analyze some distributions involving the longest and shortest negative excursions of spectrally negative Lévy processes using the binomial expansion approach. More specifically, we study the distributions of such excursions and related quantities such as the joint distribution of the shortest and longest negative excursions and their difference (also known as the range) over a random and infinite horizon time. Our results are applied to address new Parisian ruin problems, stochastic ordering and the number near-maximum distress periods showing the superiority of the binomial expansion approach for such cases.

Some New Graph Interpretations of Padovan Numbers

Padovan numbers and Perrin numbers belong to the family of numbers of the Fibonacci type and they are well described in the literature. In this paper, by studying independent (1,2)-dominating sets in paths and cycles, we obtain new binomial formulas for Padovan and Perrin numbers. As a consequence of graph interpretation, we propose a new dependence between Padovan and Perrin numbers. By studying special independent (1,2)-dominating sets in a composition of two graphs, we define Padovan polynomials of graphs. By the fact that every independent (1,2)-dominating set includes the set of leaves as a subset, in some cases a symmetric structure of the independent (1,2)-dominating set can be used.

On the non-Newtonian Padovan and non-Newtonian Perrin numbers

In this work, we introduce a novel version of Padovan and Perrin numbers which we refer to as non-Newtonian Padovan and non-Newtonian Perrin numbers. Furthermore, we examine about a number of their properties. Additionally, we provide a variety of identities and formulae involving these new kinds, including the Binet-like formulae, the generating functions, the partial sum formulae, and the binomial sum formulae.

General expressions for on-shell recursion relations for tree-level open string amplitudes

In this paper, we present a systematic derivation aimed at obtaining general expressions for on-shell recursion relations for tree-level open string amplitudes. Our approach involves applying the BCFW shift to an open string amplitude written in terms of multiple Gaussian hypergeometric functions. By employing binomial expansions, we demonstrate that the shifted amplitudes manifest simple poles, which correspond to scattering channels of intermediate states. Using the residue theorem, we thereby derive a general expression for these relations.

Transformation of surface [formula omitted]-forces in high grade elasticity

The expression of the elastic energy, postulated for a given material, specifies through the principle of virtual work which external loadings turn out to be admissible. For the elastic bodies, the energy of which depends on the gradients of the placement higher than (H+1) (or, equivalently, on the derivatives of the deformation gradient higher than H), external force densities can be prescribed over the boundary surface, which do work on the H-th derivative of the virtual displacements along the normal direction. Such generalized forces with H=1,2 have been dealt with in the literature, referred to as double and triple forces: when H is increased, however, the analytical evaluation of these work terms may become prohibitive. Each Eulerian H-force density of this kind, acting on the current boundary surface, turns out to be equivalent to a set of diverse actions defined in the Lagrangian configuration. These actions include: (i) surface terms doing work on the virtual displacement vector; (ii) surface terms doing work on the higher order derivatives of the virtual displacement along the normal direction; (iii) edge terms, doing work on the virtual displacement or on its higher order gradients, which may be further reduced. In this study, a recursive algorithm is utilized to generate all these Lagrangian contributions for any H. Such an approach rests on the binomial expansion of the tensor products of complementary surface projectors, symmetrized in the work duality, and on the integration by parts combined with the generalized divergence theorem.

Review of Vedic Mathematics Yavadunam Tavadunikritya Varganicha Yojayet Sutra for Cubing a Number

This paper discloses secrets and logics of the short formula of ancient Vedic Mathematics Yavadunam Tavadunikritya Varganicha Yojayet Sutra in finding the cube of a positive number. Through the formalization process, we have aimed to bridge the gap between ancient mathematical wisdom and modern theoretical foundations, showcasing the timeless utility of Vedic techniques in mathematical calculations. The sutra has been reviewed using the algebraic explanation in terms of decimal number system of the multiplication process by applying Binomial theorem for the positive integral exponent 3. Some more short-cut techniques have been discussed to explain the sutra and it has been found that they are producing the same result. There are shortcomings that have been found in applying the sutra, which has been solved using some other techniques as a special rule. Later the sutra has been discussed for negative integers also with a note. Before coming to the conclusion, we have discussed some limitations of the sutra, although that has been solved in the paper but with a different process. We have discussed the implications of the sutra for modern computational methods and educational practices, suggesting potential areas for further research.

0,1]Truncated Exponentiated Exponential Burr type X Distributionwith Applications

We introduce a new flexible distribution four-parameter which is called [0,1]Truncated Exponentiated Exponential Burr type X distribution ([0,1] TEE-BX). Using binomial series expansion and exponential expansion, the new distribution is expanded by four parameters. We derive moments, moment generating function, quantile function, order statistics and the Rényi entropy. The maximum likelihood estimation method is used to estimate the parameters of the proposed TEE-BX model [0,1]. Finally, using two real-world data sets, the performance of the [0,1] TEE-BX distribution is explored. Based on the certain goodness of fit criteria, we conclude that the [0,1] TEE-BX distribution has a better fit than the other distributions.

Analisis Kesalahan Mahasiswa Dalam Menyelesaikan Soal Pembuktian Teorema Binomial Pada Sifat Simetrik

This research was conducted to observe students' error in working on problems proving the binomial theorem on the symmetric properties of binomial coefficients. This type of research is qualitative research. This research uses a test method, namely by working on questions followed by interviews. This research involved students from the Tadris Mathematics Study Program at Al Falah As Sunniyah Kencong Jember University in the second semester of the 2023/2024 academic year with a total of 17 students. The 17 students were grouped into 3 cognitive abilities, namely high, medium and low cognitive. The subjects of this research consisted of three students with each student representing each cognitive ability group. The results of the research show that the types of language interpretation errors are made by subjects with high and moderate cognitive abilities, namely not writing the correct sentences when drawing conclusions. Conceptual errors were made by subjects with medium and low cognitive abilities in the form of errors in the proof steps. Procedural errors were made by subjects with moderate and low cognitive abilities, namely not having carried out complete calculations when applying the theorem in numerical form. Technical errors were made by subjects with low cognitive abilities, namely writing the same steps repeatedly. Errors in drawing conclusions were made by subjects with high, medium and low cognitive abilities. The three subjects considered the verification process to be complete. All errors are caused by a lack of understanding of the concept, carelessness in the proof process and not being careful in reading the questions.

Geometrical effective medium model of electric conduction of partially saturated clays

The current existing researches were discussed to assess the advantages and limitations of existing empirical and theoretical models for direct conductivity (DC) prediction. Prior research has demonstrated that an effective medium model may not accurately reflect the actual situation due to the presence of two types of water (bound water and bulk water) in clay-rich materials. Furthermore, the existing models can not satisfy the prediction of electrical conductivity of metal ions adsorbed clay or unsaturated clay. To address this issue, a new Effective Medium Double Water (EMDW) model was proposed based on multiple scattering techniques, which encompasses soil particles, surface-bound water layers, and bulk water and was established by controlled soil types and degrees of saturation. The novel EMDW model includes the Coherent Potential Approximation (CPA), which has consistently demonstrated superior agreement with experimental data when compared to other approximation models. Moreover, the binomial expansion approximation was utilized to simplify the formula and facilitate its use. The developed conductivity model was validated with data from other researchers. In comparison to other well-established conductivity models, the proposed EMDW model has clear physical meaning and can accurately compute matrix conductivity utilizing modified coated particle conductivity and saturation conductivity. The findings suggest that matrix conductivity in clay materials is significantly correlated with electrical-physical parameters, such as porosity, degree of saturation, shape of each discontinuous phase, and conductivity of surface-bound water and bulk water. Consequently, the new EMDW model is a theoretically grounded, physically meaningful, and easy-to-use model for conductivity prediction in clay materials.

Predicting frequency distributions of biopsied genotypes by their discrete cohort size using the binomial theorem

Predicting frequency distributions of biopsied genotypes by their discrete cohort size using the binomial theorem

Can Newtonian kinetic energy and Einsteinian rest-mass energy be expressed by the binomial expansion of the Lorentz factor? And how valid is using Einstein’s E = mc2 to calculate the nuclear fission energy?

Can Newtonian kinetic energy and Einsteinian rest-mass energy be expressed by the binomial expansion of the Lorentz factor? And how valid is using Einstein’s E = mc2 to calculate the nuclear fission energy?

Some 𝑞-identities derived by the ordinary derivative operator

In this paper, we investigate applications of the ordinary derivative operator, instead of the q q -derivative operator, to the theory of q q -series. As main results, many new summation and transformation formulas are established which are closely related to some well-known formulas such as the q q -binomial theorem, Ramanujan’s 1 ψ 1 {}_1\psi _1 formula, the quintuple product identity, Gasper’s q q -Clausen product formula, and Rogers’ 6 ϕ 5 {}_6\phi _5 formula, etc. Among these results is a finite form of the Rogers-Ramanujan identity and a short way to Eisenstein’s theorem on Lambert series.

Market consistent bid-ask option pricing under Dempster-Shafer uncertainty

We refer to the discrete-time market model under ambiguity introduced in [Cinfrignini, A., Petturiti, D. and Vantaggi, B., Dynamic bid–ask pricing under Dempster-Shafer uncertainty. J. Math. Econ., 2023a, 107, 102871], formed by a frictionless risk-free bond and a non-dividend paying stock with bid-ask spread. For a European derivative, we generalize the classical binomial pricing formula by allowing for bid-ask prices and investigate the properties of the ensuing replicating strategies. Next, for an American derivative, we propose a backward bid-ask pricing procedure and prove that the resulting discounted price processes are the bid-ask Choquet-Snell envelopes of the discounted payoff process, respectively. Moreover, for an American call option, we prove a generalization of the well-known Merton's theorem [Merton, R.C., Theory of rational option pricing. Bell J. Econ. Manage. Sci., 1973, 4, 141–183] holding for both the bid and the ask price processes. Finally, we introduce a market consistent calibration procedure and show the use of the calibrated model in bid-ask option pricing.

Accurate analytical evaluation of the generalized logarithmic and double Fermi–Dirac and Bose–Einstein functions

AbstractThe accurate definition and powerful evaluation modeling of the various generalized Fermi–Dirac and Bose–Einstein functions remain a challenging problem in various areas of physics. In this study, we develop a general analytical technique for accurately calculating logarithmic and double Fermi–Dirac and Bose–Einstein functions. The obtaining analytical formulae are established by considering the binomial expansion theorem. The obtained expressions are valid in chemical potential values between ‐∞ <μ <0 and have been designated as explicit form features, high precision, and less computing time. The calculation results are tabularly illustrated to show the consistency of the analytical relations analysis under the effect of parameters. Based on a comprehensive analysis of the results, they are potentially useful in applications to evaluate thermionic emission and astrophysics problems.

Series expansion of some elementary functions without using mathematical analysis

We present a method to compute the power series expansions of e x , ln ⁡ ( 1 + x ) , sin ⁡ x , and cos ⁡ x without relying on mathematical analysis. Using the properties of elementary functions, we determine the coefficients of each series through the method of undetermined coefficients. We have validated our formulae through the use of mathematical induction. The Newton binomial formula is obtained using the expansions of the exponent and the logarithm in their power series. We build upon the approach outlined in a paper by L. P. Mironenko and O. A. Rubtsova (Approximations of Some Functions by Polynomials and the Method of Undefined Coefficients, Scientific Papers of Donetsk National Technical University. Series: Computing and Automation, No. 2 (25), p. 128–135, 2013) and complement their approach. This approach does not depend on the application of Taylor's theorem to the expansion of functions. Consequently, these expansions can be integrated into the educational process before students become acquainted with the concept of derivatives. It simplifies the study of the theory of limits, especially in the computation of standard limits such as lim x → 0 sin ⁡ ( x ) x , lim x → 0 1 − cos ⁡ ( x ) x 2 , and lim x → 0 ( 1 + x ) 1 / x .

Generalization of the Fibonacci sequence, Pascal's triangle, and the binomial theorem

هدف هذا البحث إلى تعميم العلاقة بين مثلث باسكال وتوسيع ثنائي باستخدام المتغيرات بدلاً من الأرقام. يتم تشكيل المثلث باستخدام المتغير (d) بدلاً من المصطلح الصفري (0)، والمتغير (a) بدلاً من المصطلح الأول (1)، والمتغير (m) كتعميم لنظرية الثنائي. يتم دراسة الأنماط الرياضية الناتجة عن تشكيل المثلث باستخدام هذه المتغيرات، مما يؤدي إلى خمسة معادلات رياضية جديدة: معادلة عمودية، معادلة الوتر، معادلة الصف، مجموع معادلات الصفوف، ومعادلة تسلسلات الوظائف الذهبية. تُعتبر معادلة تسلسلات الوظائف الذهبية تعميمًا غير مسبوق للمصطلح الثاني لتسلسل فيبوناتشي وتسلسل لوكاس. بالإضافة إلى ذلك، يُصاغ معادلة جديدة وغير مسبوقة للماس، ويُصاغ افتراض جديد يتعلق بأعداد أولية، حيث يُعتبر تعميمًا لنظرية فيرما الصغرى. يُسلط هذا البحث الضوء على ضرورة فهم أكثر شمولًا للعلاقة بين مثلث باسكال وتوسيع ثنائي.

Efficient LiDAR-Trajectory Affinity Model for Autonomous Vehicle Orchestration

Computation and memory resource management strategies are the backbone of continuous object tracking in intelligent vehicle orchestration. Multi-object tracking generates enormous measurements of targets and extended object positions using light detection and ranging (Lidar) sensors. Designing an adequate object-tracking system is a global challenge because of dynamic object detection and data association uncertainties during scene understanding. In this regard, we develop an intelligent multi-objective tracking (IMOT) system with a novel measurement model, called the box data association inflate (BDAI) model, to assess each target’s object state and trajectory without noise by using the Bayesian approach. The box object filter method filters ambiguous detection responses during data association. The theoretical proof of the box object filter is derived based on binomial expansion. Prognosticating a lower-dimension object than the original point object reduces the computational complexity of vehicle orchestration. Two datasets (NuScenes dataset and our lab dataset) are considered during the simulations, and our approach measures the kinematic states adequately with reduced computation complexity compared to state-of-the-art methods. The simulation outcomes show that our proposed method is effective and works well to detect and track objects. The NuScenes dataset contains 28130 samples for training, 6019 examples for validation and 6008 samples for testing. IMOT achieves 58.09% tracking accuracy and 71% mAP with 5 ms pre-processing time. The Jetson Xavier NX consumes 49.63% GPU and 9.37% average power and exhibits 25.32 ms latency compared to other approaches. Our system trains a single pair frame in 169.71 ms with affinity estimation time of 12.19 ms, track association time of 0.19 ms and mATE of 0.245 compared to state-of-the-art approaches

On Computing General Root Algorithms Based on Binomial Expansion, Bernoulli’s Method of Continuous Compound Interest, Series Expansion and Other Modification Methods

Most of the Pth root algorithms exhibit high latency or small convergence rates when iteratively computing the roots. Here we present a slew of algorithms based on series expansions of binomial form, and exponential terms that show low latency or small computational cost for finding the Pth root. The latency decreases if the family of series taken are truncated at higher terms. We show that Babylonian method converges quadratically while the binomial series show an increase in convergence rate from quadratic, cubic and higher order O(t^N) convergence rate as the series is sequentially truncated at higher order terms.

Sun’s Binomial Inversion Formulae, Euler’s Operator, and Z-Transform

We exhibit the participation of Euler’s operator in the Sun’s binomial inversion formulae and we use the Z-transform to give elementary proofs of such formulae.

Binom Katsayılı Geometrik Serilere Farklı Bir Bakış

The study of mathematical series has long been a fascinating and essential component of mathematics, providing valuable insights into numerous real-world applications and theoretical concepts. Among the various types of series, the "Geometric Series with Binomial Coefficients" stands out as a particularly intriguing and powerful subject of investigation. 
 
 A geometric series is a sequence of terms in which each successive term is obtained by multiplying the previous one by a constant factor, known as the common ratio. This classical concept has found extensive applications in fields like finance, physics, engineering, and computer science, making it an indispensable tool for solving a wide array of problems. 
 
 However, in the context of the "Geometric Series with Binomial Coefficients," we encounter a fascinating twist that elevates the complexity and versatility of the series. Instead of dealing with constant factors as in the traditional geometric series, the coefficients in this new variant are given by the binomial coefficient formula. Binomial coefficients, also known as "n choose k," are fundamental in combinatorial mathematics and represent the number of ways to choose k elements from a set of n elements. 
 
 This work presents a new approach for the computation to geometric series with binomial coefficients. The geometric series with binomial coefficients is derived from the multiple summations of a geometric series. In this article, several theorems and corollaries are established on the innovative geometric series and its binomial coefficients.