Nonlinear Inverse Relations of the Bell Polynomials via the Lagrange Inversion Formula (Ⅱ)

In this paper,by means of the classical Lagrange inversion formula,the authors establish a general nonlinear inverse relation as the solution to the problem proposed in the paper[J.Wang,Nonlinear inverse relations for the Bell polynomials via the Lagrange inversion formula,J.Integer Seq.,Vol.22(2019),Article 19.3.8].As applications of this inverse relation,the authors not only find a short proof of another nonlinear inverse relation due to Birmajer,et al.(2012),but also set up a few convolution identities concerning the Mina polynomials.

Knotted Picture of Unitary Transformation Applied to Single Particle Quantum State

This paper introduces the unified expression of four Bell bases and uses the composition operation in knot theory to obtain the knotted picture of unitary operators based on Pauli operators, and gives the knotted expressions of 12 inversion relations between the four Bell bases.

Correspondence Between Bell Bases and Oriented Links in Knot Theory

From the comparison of correlation tensor in the theory of quantum network, the Alexander relation matrix in the theory of knot crystals and the identical inversion relations under the action of Pauli matrices, we show that there is a one to one correspondence between four Bell bases and four oriented links of the linkage in knot theory.

A natural chemical technology for reducing fluoridation through chloride contents in muscles of fish

Accumulation trend of fluoride ion was carried out in 25 fish species, collected randomly from fresh water and marine water, Sindh, Pakistan to ensure the level of chloride contents by Selective Ion Electrode （SIE） method. Results showed an inverse relation in between chloride and fluoride concentration in muscles of fish under investigation, It was observed that chloride ions contents of marine water fish were significantly higher as compared with fresh water fish. This may be related with the habitat of marine water fish while no appreciable divergence in fluoride concentration in both regions was observed. This indicated that the deliberation of fluoride is the phenomena of nature of fish and self-determining. Highest concentration of fluoride ion in fresh water fish and marine water fish showed an inverse level of chloride ion in it, reflects water body conditions of both resources. A probable mechanism of accumulation of fluoride with respect to essential chloride ion which may be related with the size of ionic radii of both halides ion is described in relevant section of this article. Where a larger ionic radii of chloride ion believed to play an imperative role in reducing fluoridation in fishes. This may be regarded as natural chemical technology for reducing fluoride in muscle （edible part） of fish of both regions.

Estimation of the water–oil–gas relative permeability curve from immiscible WAG coreflood experiments using the cubic B-spline model

Immiscible water-alternating-gas（WAG） flooding is an EOR technique that has proven successful for water drive reservoirs due to its ability to improve displacement and sweep efficiency.Nevertheless,considering the complicated phase behavior and various multiphase flow characteristics,gas tends to break through early in production wells in heterogeneous formations because of overriding,fingering,and channeling,which may result in unfavorable recovery performance.On the basis of phase behavior studies,minimum miscibility pressure measurements,and immiscible WAG coreflood experiments,the cubic B-spline model（CBM） was employed to describe the three-phase relative permeability curve.Using the Levenberg-Marquardt algorithm to adjust the vector of unknown model parameters of the CBM sequentially,optimization of production performance including pressure drop,water cut,and the cumulative gas-oil ratio was performed.A novel numerical inversion method was established for estimation of the water-oil-gas relative permeability curve during the immiscible WAG process.Based on the quantitative characterization of major recovery mechanisms,the proposed method was validated by interpreting coreflood data of the immiscible WAG experiment.The proposed method is reliable and can meet engineering requirements.It provides a basic calculation theory for implicit estimation of oil-water-gas relative permeability curve.

A Formula Containing Four Independent Bases and Its Applications

A transformation formula containing four independent bases is found by a special inversion formula and it is applied to derive a few summation formulas for basic hypergeometric series only by elementary method. The hypergeometric series, the limits of those formulas are also obtained.

A Short Proof of Krattenthaler Formulas

With an effort to investigate a unified approach to the Lagrange inverse Krattenthaler established operator method we finally found a general pair of inverse relations,called the Krattenthaler formulas.The present paper presents a very short proof of this formula via Lagrange interpolation. Further.our method of proof declares that the Krattenthaler result is unique in the light of Lagrange interpolation.