Binomial Theorem Research Articles

Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. II. Non-integer orders

This paper constitutes the second part on the subject of finite part integration of the generalized Stieltjes transform Sλ[f]=∫0∞f(x)(ω+x)−λdx about ω = 0, where now λ is a non-integer positive real number. Divergent integrals with singularities at the origin are induced by writing (ω + x)−λ as a binomial expansion about ω = 0 and interchanging the order of operations of integration and summation. The prescription of finite part integration is then implemented by interpreting these divergent integrals as finite part integrals which are rigorously represented as complex contour integrals. The same contour is then used to express Sλ[f] itself as a complex contour integral. This led to the recovery of the terms missed by naïve term-wise integration which themselves are finite parts of divergent integrals whose singularity is at the finite upper limit of integration. When the function f(x) has a zero at the origin of order m = 0, 1, … such that m − λ < 0, the correction terms missed out by the naïve term by term integration give the dominant contribution to Sλ[f] as ω → 0. Otherwise, the correction term is sub-dominant to the leading convergent terms in the naïve term by term integration. We apply these results by obtaining exact and asymptotic representations of the Kummer and Gauss hypergeometric functions by evaluating their known Stieltjes integral representations. We then apply the method of finite part integration to obtain the asymptotic behavior of a generalization of the Stieltjes integral which is relevant in the calculation of the effective index of refraction of a shallow potential well.

Research on Reflection and Rotation Features of Binomial Coefficient Distributions

With the rapid development of frontier neural network, artificial intelligence and big data technology, the related theory and application research of combinatorial mathematics are more and more extensive. The application of binomial coefficient in combinatorial mathematics is an effective way to solve the research of information coding and quantum computation. Binomial coefficients and its distributions are the core topic in probability statistics, and there are many theories and applications related to them.In this paper, the quantification of binomial coefficients and the characteristic of reflection rotation transformation are studied using three-dimensional diagrams. Using variant construction, the combinatorial clustering properties are investigated applying binomial formulas and sample distributions and their combinatorial patterns are illustrated. It is proved that the basic binomial coefficient formula and its extended model have obvious properties of reflection and rotation invariance.

Prevalence of subtypes of multiple sclerosis and the most common clinical symptoms in Iranian patients: A meta‐analysis

AbstractMultiple sclerosis (MS) is an inflammatory disease defined by damage to the myelin sheaths of neurons. MS is classified into different subtypes including relapsing–remitting MS, secondary progressive MS and primary progressive MS. The present study aimed to estimate the prevalence of different forms and clinical symptoms of MS among Iranian people using meta‐analysis. We searched all available databanks, such as SID, Magiran, PubMed, Google scholar, Scopus and Science, directly to identify relevant evidence. All findings were reviewed and eligible articles were selected. The standard error of prevalence was calculated based on binomial distribution formula. The heterogeneity between the results of primary studies was detected using Cochrane (Q) and I2 indices. The primary results were combined according to a random effects model. The effect of related factors in heterogeneity was investigated using a meta‐regression model. During the primary search, 17 352 articles were identified. After limiting the search strategy, and excluding duplicates and irrelevant papers, finally, 21 primary studies were entered into the final meta‐analysis. The total prevalences of relapsing–remitting MS, primary progressive MS and secondary progressive MS types among Iranian people were estimated to be 69.7% (95% CI 63.7–75.8), 7.01% (95% CI 4.05–9.96) and 17.8% (95% CI 11.7–23.9) respectively. In addition, prevalences of fatigue and motor dysfunction were estimated to be 71.1% (95% CI 60.9–81.3) and 56.3% (95% CI 27.9–84.6), respectively. Our meta‐analysis showed that relapsing–remitting MS is the most common type of MS, and fatigue is the most common clinical manifestation in patients. Motor dysfunction and balance disorder are the other common problems in patients.

A Simple Design of Fractional Delay FIR Filter Based on Binomial Series Expansion Theory

Fractional delay filters modeling non-integer delays are digital filters that ideally have flat group delays. This paper proposes a simple design method of fractional delay FIR filter based on binomial series expansion theory. First, the design technique is based on the binomial series expansion method which is applied to a discrete fractional system to obtain a closed form FIR digital filter which approximates the digital fractional delay operator z−m $$ (m \in \Re^{ + } ) $$ . Then, the principal differentiation is used to design fractional delay FIR filter with a broader group delay bandwidth. Finally, numerical examples of fractional delay FIR filter design show that the proposed approach yields better performance compared to the existing techniques.

Object Reconstruction Using the Binomial Theorem for Ghost Imaging

Noise term in the reconstruction matrix in ghost imaging is a major cause of blurring imaging results. To remedy this problem, we propose a new ghost imaging method based on the binomial theorem to reduce the level of noise. In our method, images with low-level noise can be generated by constructing a binomial formula using high-order imaging results that are acquired by reintroducing the reconstruction result back into the imaging formula repeatedly. Experimental and simulation results demonstrate that our method is effective in improving imaging quality and the anti-interference performance and reducing computing time, making it useful for practical applications.

Arithmetic properties of bipartitions with ℓ-core

Let Aℓ(n) denote the number of bipartitions of n with ℓ-core. In this paper, we prove a theorem on congruences modulo 2 for Aℓ(n) with ℓ≥2 by employing a well known congruence due to binomial theorem. Also, we show some interesting congruences for some special ℓ’s.

Analytical calculation of stress intensity factors for cracks emanating from a quasi-square hole in an infinite plane

A new conformal mapping is proposed in order to solve stress intensity factor (SIF) problem in an infinite plane containing a traction-free square hole with two unequal cracks, analytically. To this end, Schwarts-Christoffel integral is expanded as sum of finite term series through Newton’s binomial formula to approximate the square hole to a rounded corner square one, which is called quasi-square. Next, the mapping function of cracked hole is expanded to a series of fractional expressions to combine with Muskhelishvili formulation. Finally, SIFs are presented for the remote tensile loading and the effects of crack length and radius of corners on the SIF are studied. To ensure the accuracy, results are compared to the available literature and FEM. Contrary to the previous researches, it was shown that the hole-shape had a considerable effect on the SIF for the small cracks though for large length cracks it is negligible.

Selective detections of single-viruses using solid-state nanopores

Rapid diagnosis of flu before symptom onsets can revolutionize our health through diminishing a risk for serious complication as well as preventing infectious disease outbreak. Sensor sensitivity and selectivity are key to accomplish this goal as the number of virus is quite small at the early stage of infection. Here we report on label-free electrical diagnostics of influenza based on nanopore analytics that distinguishes individual virions by their distinct physical features. We accomplish selective resistive-pulse sensing of single flu virus having negative surface charges in a physiological media by exploiting electroosmotic flow to filter contaminants at the Si3N4 pore orifice. We demonstrate identifications of allotypes with 68% accuracy at the single-virus level via pattern classifications of the ionic current signatures. We also show that this discriminability becomes >95% under a binomial distribution theorem by ensembling the pulse data of >20 virions. This simple mechanism is versatile for point-of-care tests of a wide range of flu types.

The Bregman-divergence universal portfolio associated with a convex polynomial

A convex polynomial with a non-integer degree is used to generate a Bregman- divergence universal portfolio together with an application of a first-order binomial expansion of the multiplicative update. The next-day portfolio is a weighted combination of the present- day portfolio and a power of the present-day portfolio, where the weights are depending on functions of the present-day stock price relatives. A parameter delta is introduced in the portfolio with aims to find the parameter that exhibiting the best performance locally. The portfolio is run over five stock data sets selected from the local stock exchange to study and to compare its empirical performance. The empirical results are compared with the results obtained by using the reverse Helmbold universal portfolio. Better performance is demonstrated for the portfolio introduced in this paper with respect to certain values of the parameters used on three data sets. There is promising evidence that this portfolio can increase the investment wealth significantly with the proper choice of the portfolio parameters.

ANALYTICAL EVALUATION OF THE EINSTEIN INTEGRATE USING BINOMIAL EXPANSION THEOREM AND POWER SERIES

The objective of this paper is to provide an efficient and reliable analytical expression for the Einstein integrals using the binomial expansion theorem and power series. The obtained analytical expressions are valid for all values of their arguments. The algorithm can be used in the software and simulation programs. Furthermore, the comparison of the method with numerical calculations shows the applicability and accuracy of the method.

The Binomial Theorem and motivic classes of universal quasi-split tori

Taking symmetric powers of varieties can be seen as a functor from the category of varieties to the category of varieties with an action by the symmetric group. We study a corresponding map between the Grothendieck groups of these categories. In particular, we derive a binomial formula and use it to give explicit expressions for the classes of universal quasi-split tori in the equivariant Grothendieck group of varieties.

A Review of Multiple Approaches for Binomial Theorem

A Review of Multiple Approaches for Binomial Theorem

Analysis of the Generalized Inverse Polynomial Scheme to Initial Value Problems

In this paper, we study the analysis of the generalized inverse polynomial scheme for the numerical solution of initial value problems of ordinary differential equation. At first, we generalize the scheme up to the fifth stage using the Binomial expansion and Taylor’s series method towards its derivation. The trend shows the generalization to the kth term. The analysis demonstrates the efficiency and the effectiveness of the generalized scheme.Keywords— Taylor’s Series, Initial Value Problem, Stability, Consistency, Convergence.

Three new classes of generalized almost perfect nonlinear power functions

Generalized almost perfect nonlinear (GAPN) functions are algebraic generalization of almost perfect nonlinear functions and have important applications in cryptography and finite geometry. Based on the well known binomial theorem and some combinatory formulas, the conjecture proposed by Kuroda is partially disproved and three new classes of GAPN power functions for odd characteristic are presented.

Uniform symbolic topologies via multinomial expansions

When does a Noetherian commutative ring R R have uniform symbolic topologies on primes–read, when does there exist an integer D > 0 D>0 such that the symbolic power P ( D r ) ⊆ P r P^{(Dr)} \subseteq P^r for all prime ideals P ⊆ R P \subseteq R and all r > 0 r >0 ? Groundbreaking work of Ein-Lazarsfeld-Smith, as extended by Hochster and Huneke, and by Ma and Schwede in turn, provides a beautiful answer in the setting of finite-dimensional excellent regular rings. It is natural to then search for analogues where the ring R R is non-regular, or where the above ideal containments can be improved using a linear function whose growth rate is slower. This manuscript falls under the overlap of these research directions. Working with a prescribed type of prime ideal Q Q inside of tensor products of domains of finite type over an algebraically closed field F \mathbb {F} , we present binomial and multinomial expansion criteria for containments of type Q ( E r ) ⊆ Q r Q^{(E r)} \subseteq Q^r , or even better, of type Q ( E ( r − 1 ) + 1 ) ⊆ Q r Q^{(E (r-1)+1)} \subseteq Q^r for all r > 0 r>0 . The final section consolidates remarks on how often we can utilize these criteria, presenting an example.

Weight-dependent commutation relations and combinatorial identities

We derive combinatorial identities for variables satisfying specific systems of commutation relations, in particular elliptic commutation relations. The identities thus obtained extend corresponding ones for q-commuting variables x and y satisfying yx=qxy. In particular, we obtain weight-dependent binomial theorems, functional equations for generalized exponential functions, we derive results for an elliptic derivative of elliptic commuting variables, and finally study weight-dependent extensions of the Weyl algebra which we connect to rook theory.

Some results for the irreducibility of truncated binomial expansions

For positive integers k and n with k⩽n−1, let Pn,k(x) denote the polynomial ∑j=0k(nj)xj, where (nj)=n!j!(n−j)!. In 2011, Khanduja, Khassa and Laishram proved the irreducibility of Pn,k(x) over the field Q of rational numbers for those n,k for which 2≤2k≤n<(k+1)3. In this paper, we extend the above result and prove that if 2≤2k≤n<(k+1)e+1 for some positive integer e and the smallest prime factor of k is greater than e, then there exists an explicitly constructible constant Ce depending only on e such that the polynomial Pn,k(x) is irreducible over Q for k≥Ce.

Efficacy of Hepatitis B Vaccination in Under Five-Year-Old Children in Iran: A Systematic Review and Meta-Analysis Study

Context: Hepatitis B viral infection, specially in children, is an unsolved global health issue. National and international vaccination programs, as the main means to prevent this infection, have been operating for years. This meta-analysis study was launched to investigate the vaccination efficacy in under five-year-old children in Iran. Evidence Acquisition: In this research, all national databases including, SID, Magiran, Iran medex, Iran doc, and Medlib, as well as international databases, including PubMed, Medline, and ISI were searched for published articles associated with evaluation of seroconversion rates after hepatitis-B vaccination in Iranian children. The search was performed in the Persian and/or English language and in under five-year-old children in Iran. The variance of each study was calculated using the binomial distribution formula. The results of studies were combined using random-effects meta-analysis model. Data were analyzed using STATA version 12. Results: In the present research, 11 studies, performed from 2000 to 2017, were investigated. The number of samples in these 11 studies were 3063 children (1592 males and 1471 females). In the studies, the efficacy range of the hepatitis-B vaccine in under five-year-old children in Iran was 89 (95 CI: 86 -93) with high heterogeneity (P value for heterogeneity was 0.001, I-2 = 92.9). Range of efficacy for males and females was 85 (95 CI: 78 to 91) and 88 (95 CI: 83 to 93), respectively. Conclusions: The hepatitis B virus (HBV) vaccination program in Iran seems to be highly effective although some questions, such as the effect of gender on responsiveness to vaccine and heterogeneity of different data, remain unclear.

HSIC-based kernel independent component analysis for fault monitoring

For nonlinear and non-Gaussian industrial processes, KICA is a mature and successful method for fault monitoring. However, a fixed-point ICA algorithm uses a negative entropy method, which has relatively high requirements for non-quadratic function and initial point; accuracy is unsatisfactory. The calculation of control limits of Hotelling's T2 statistic is based on a kernel density estimation, which is difficult to calculate and implement and sometimes cannot guarantee accuracy. In this paper, the kernel independent component analysis method based on the Hibert-Schmidt independence criterion (HSIC) is used instead of the fixed-point ICA algorithm for fault monitoring to improve the accuracy of the independent element. We obtain the independent element directly from the objective function, rather than through the combination of KPCA and ICA. At the same time, we use the direct binomial expansion theorem to obtain the control limit, which reduces computational complexity and implementation difficulty and improves accuracy. The control limit is improved to obtain multi-fault diagnosis. Gray-level information and color information of each frame of the video are respectively read through the information entropy and HSV spatial color histogram. Experimental results show the advantage and effectiveness of the proposed approach. Meanwhile, shadow variables are introduced to smooth the statistics. The contributions of this paper are as follows. 1) Using the kernel independent component method based on HSIC for fault monitoring improves speed and accuracy. 2) The binomial expansion theorem is used instead of traditional kernel density estimation to calculate the control limit, which improves the results of fault monitoring. 3) A method of fault detection using information entropy, HSV color histogram and multivariate statistical analysis is presented.

Excursion Processes Associated with Elliptic Combinatorics

Researching elliptic analogues for equalities and formulas is a new trend in enumerative combinatorics which has followed the previous trend of studying $q$-analogues. Recently Schlosser proposed a lattice path model in the square lattice with a family of totally elliptic weight-functions including several complex parameters and discussed an elliptic extension of the binomial theorem. In the present paper, we introduce a family of discrete-time excursion processes on $\mathbb{Z}$ starting from the origin and returning to the origin in a given time duration $2T$ associated with Schlosser's elliptic combinatorics. The processes are inhomogeneous both in space and time and hence expected to provide new models in non-equilibrium statistical mechanics. By numerical calculation we show that the maximum likelihood trajectories on the spatio-temporal plane of the elliptic excursion processes and of their reduced trigonometric versions are not straight lines in general but are nontrivially curved depending on parameters. We analyze asymptotic probability laws in the long-term limit $T \to \infty$ for a simplified trigonometric version of excursion process. Emergence of nontrivial curves of trajectories in a large scale of space and time from the elementary elliptic weight-functions exhibits a new aspect of elliptic combinatorics.