Binomial coefficient

In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written \tbinom{n}{k}. It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n; this coefficient can be computed by the multiplicative formula :\binom nk = \frac{n\times(n-1)\times\cdots\times(n-k+1)}{k\times(k-1)\times\cdots\times1}, which using factorial notation can be compactly expressed as :\binom{n}{k} = \frac{n!}{k! (n-k)!}. For example, the fourth power of 1 + x is :\begin{align} (1 + x)^4 &= \tbinom{4}{0} x^0 + \tbinom{4}{1} x^1 + \tbinom{4}{2} x^2 + \tbinom{4}{3} x^3 + \tbinom{4}{4} x^4 \ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end{align} and the binomial coefficient \tbinom{4}{2} =\tfrac{4\times 3}{2\times1} = \tfrac{4!}{2!2!} = 6 is the coefficient of the x2 term. Arrangi

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Nouvelle méthode de mesure directe de l'effet piézooptique; application au quartz

The effect of an elastic deformation on optical properties of an aeolotropic medium (piezo-optic effect) is explicitly established from an energy function. Then is deduced the number of distinct coefficients ruling this effect in the special case of quartz. The author gives a new method to measure the absolute dephasage ; method based on photometric exploration of the three beams diffracted image, around the focus. By this method a phase variation of 2π/2000 has been disclosed. This method has been used for the direct measure of plaza-optic effect in quartz. With a single type of cubic sample, 5 out of the 8 coefficients of quartz have been accurately settled, and an upper limit has been given for the one of others. Other applications of this measurement method are considered.

s.n.1956

An algorithm for computing weight multiplicities in irreducible modules for complex semisimple Lie algebras

Mikaël Cavallin

Let be a finite-dimensional semisimple Lie algebra over having rank l and let V be an irreducible finite-dimensional -module having highest weight λ. Computations of weight multiplicities in V, usually based on Freudenthal's formula, are in general difficult to carry out in large ranks or for λ with large coefficients (in terms of the fundamental weights). In this paper, we first show that in some situations, these coefficients can be “lowered” in order to simplify the calculations. We then investigate how this can be used to improve the aforementioned formula of Freudenthal, leading to a more efficient version of the latter in terms of complexity as well as to a way of dealing with certain computations in unbounded ranks. We conclude by illustrating the last assertion with a concrete example.

2017

Quantum state resolved molecular beam reflectivity measurements: CH4 dissociation on Pt(111)

Rainer Beck, Helen Jane Chadwick, Ana Gutiérrez González

The King and Wells molecular beam reflectivity method has been used for a quantum state resolved study of the dissociative chemisorption of CH4 on Pt(111) at several surface temperatures. Initial sticking coefficients were measured for incident CH4 prepared both with a single quantum of antisymmetric stretch vibration by infrared laser pumping, and without laser excitation. Vibrational excitation of the mode is observed to be less efficient than incident translational energy in promoting the dissociation reaction with a vibrational efficacy = 0.65. The initial state resolved sticking coefficient was found to be independent of the surface temperature over the 50kJ/mol to 120kJ/mol translational energy range studied here. However, the surface temperature dependence of the King and Wells data reveals the migration of adsorbed carbon formed by CH4 dissociation on the Pt(111) surface leading to the growth of carbon particles.

American Institute of Physics2016

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In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a formal power series. This series is called the generating fu

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In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x + y

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In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers le

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