Note on the evaluation method of the configurational partition function of a polypeptide molecule: Relationship between the matrix method and the generating function method

1996 ◽  
Vol 104 (16) ◽  
pp. 6405-6406
Author(s):  
Yukio Kobayashi
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Matrix Method ◽  
Function Method ◽  
Evaluation Method ◽  
The Matrix

scholarly journals Combinatorics of the Three-Parameter PASEP Partition Function

10.37236/509 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Matthieu Josuat-Vergès
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Exclusion Process ◽  
Asymmetric Exclusion Process ◽  
Combinatorial Interpretation ◽  
Rook Placements ◽  
Combinatorial Interpretations ◽  
The Matrix ◽  
The One ◽  
New Formula

We consider a partially asymmetric exclusion process (PASEP) on a finite number of sites with open and directed boundary conditions. Its partition function was calculated by Blythe, Evans, Colaiori, and Essler. It is known to be a generating function of permutation tableaux by the combinatorial interpretation of Corteel and Williams. We prove bijectively two new combinatorial interpretations. The first one is in terms of weighted Motzkin paths called Laguerre histories and is obtained by refining a bijection of Foata and Zeilberger. Secondly we show that this partition function is the generating function of permutations with respect to right-to-left minima, right-to-left maxima, ascents, and 31-2 patterns, by refining a bijection of Françon and Viennot. Then we give a new formula for the partition function which generalizes the one of Blythe & al. It is proved in two combinatorial ways. The first proof is an enumeration of lattice paths which are known to be a solution of the Matrix Ansatz of Derrida & al. The second proof relies on a previous enumeration of rook placements, which appear in the combinatorial interpretation of a related normal ordering problem. We also obtain a closed formula for the moments of Al-Salam-Chihara polynomials.

scholarly journals Matrix Ansatz, lattice paths and rook placements

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
S. Corteel ◽  
M. Josuat-Vergès ◽  
T. Prellberg ◽  
M. Rubey
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Laguerre Polynomials ◽  
Lattice Paths ◽  
Rook Placements ◽  
Combinatorial Interpretations ◽  
The Matrix ◽  
International Audience ◽  
Enumeration Formula

International audience We give two combinatorial interpretations of the Matrix Ansatz of the PASEP in terms of lattice paths and rook placements. This gives two (mostly) combinatorial proofs of a new enumeration formula for the partition function of the PASEP. Besides other interpretations, this formula gives the generating function for permutations of a given size with respect to the number of ascents and occurrences of the pattern $13-2$, the generating function according to weak exceedances and crossings, and the $n^{\mathrm{th}}$ moment of certain $q$-Laguerre polynomials. Nous donnons deux interprétations combinatoires du Matrix Ansatz du PASEP en termes de chemins et de placements de tours. Cela donne deux preuves (presque) combinatoires d'une nouvelle formule pour la fonction de partition du PASEP. Cette formule donne aussi par exemple la fonction génératrice des permutations de taille donnée par rapport au nombre de montées et d'occurrences du motif $13-2$, la fonction génératrice par rapport au nombre d'excédences faibles et de croisements, et le $n^{\mathrm{ième}}$ moment de certains polynômes de $q$-Laguerre.

Determination of fiber volume fraction in a cured laminate

2021 ◽  
pp. 002199832110112
Author(s):  
Qing Yang Steve Wu ◽  
Nan Zhang ◽  
Weng Heng Liew ◽  
Vincent Lim ◽  
Xiping Ni ◽  
...  
Keyword(s):  
Carbon Fiber ◽  
Fiber Volume Fraction ◽  
Evaluation Method ◽  
Volume Fraction ◽  
Matrix Material ◽  
Volume Ratio ◽  
Gravimetric Analysis ◽  
Fiber Volume ◽  
Laser Ultrasonic ◽  
The Matrix

Propagation of ultrasonic wave in Carbon Fiber Reinforced Polymer (CFRP) is greatly influenced by the material’s matrix, resins and fiber volume ratio. Laser ultrasonic broadband spectral technique has been demonstrated for porosity and fiber volume ratio extraction on unidirection aligned CFRP laminates. Porosity in the matrix materials can be calculated by longitudinal wave attenuation and accurate fiber volume ratio can be derived by combined velocity through the high strength carbon fiber and the matrix material with further consideration of porosity effects. The results have been benchmarked by pulse-echo ultrasonic tests, gas pycnometer and thermal gravimetric analysis (TGA). The potentials and advantages of the laser ultrasonic technique as a non-destructive evaluation method for CFRP carbon fiber volume fraction evaluation were demonstrated.

Determination of the Force Constants of Non-linear XY2 Molecules in the Gas-Phase by the GF Matrix Method

2004 ◽  
Vol 59 (9) ◽  
pp. 621-622 ◽  
Author(s):  
Fatih Ucun ◽  
Vesile Gūçlü
Keyword(s):  
Gas Phase ◽  
Condensed Phase ◽  
Matrix Method ◽  
Solid Phase ◽  
Force Constants ◽  
Matrix Solution ◽  
The Matrix ◽  
Newton Raphson ◽  
Raphson Method

The force constants of the internal coordinates of nonlinear XY2 molecules in the gas-phase were calculated by using the GF matrix method. The matrix solution was carried out by means a computer program built relative to the Newton-Raphson method and the calculations were listed in a table. The force constants of some molecules in the liquidand solid- phase were also found and compared with these ones, and it was seen that the force constants for more condensed phase are lower as in an agreement with having its lower frequency.

scholarly journals Multivariate analytical methods for simultaneous estimation of Atenolol and Hydrochlorothiazide in bulk and tablet dosage form

2021 ◽  
Vol 6 (4) ◽  
pp. 205-208
Author(s):  
Sreenivasa Charan Archakam ◽  
Keerthisikha Palur ◽  
Praveen Kumar Arava
Keyword(s):  
Linear Regression ◽  
Matrix Method ◽  
Linear Regression Analysis ◽  
Molar Absorptivity ◽  
Simultaneous Estimation ◽  
Linear Regression Method ◽  
Spectrophotometric Methods ◽  
The Matrix ◽  
Ft Ir ◽  
Tablet Dosage

The present study aimed to develop simple, accurate and precise FTIR and UV spectrophotometric methods for the quantification of Atenolol and Hydrochlorothiazide in bulk and tablet dosage forms. FT-IR method like classical least squares (CLS) was developed within the range of 2366.69-3433.44; 1564.40-1673.30 cm- UV methods like Cramer’s matrix method (method-I) and linear regression analysis (Method II) were developed and they are based upon constructing the matrix set by using molar absorptivity values at 275.60 nm and 270.40 nm. The assay values for FTIR- CLS method were 102% and 108 % for Atenololand Hydrochlorothiazide respectively. Cramer’s matrix method results were found to be 95.15% and 104% for Atenolol and Hydrochlorothiazide respectively and for linear regression method they were found to be 98.50% and 106% (w/w).

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