Incomplete Beta Research Articles

Effect of Deposition Conditions on Intrinsic Stress, Phase Transformation and Stress Relaxation in Tantalum Thin Films

AbstractWe demonstrate that the high temperature polymorphic tantalum phase transition from the tetragonal beta phase to the cubic alpha phase causes complete stress relaxation and a large decrease in the resistance of tantalum thin films. 100 nm beta tantalum thin films were deposited onto thermally oxidized <100> silicon wafers by dc magnetron sputtering with argon. In situ stress and resistance at temperature were measured during temperature-ramped annealing in purified He. Upon heating, films that were initially compressively stressed showed increasing compressive stress due to thermo-elastic deformation from 25 to 550°C, slight stress relief due to plastic deformation from 550 to 700°C and complete stress relief due to the beta to alpha phase transformation at approximately 700–800°C. Incomplete compressive stress relaxation was observed at high temperatures if the film was initially deposited in the alpha phase or if the beta phase did not completely transform into alpha by 800°C. This incomplete beta to alpha phase transition was most commonly observed on samples that had radio frequency substrate bias greater than -100 V. We conclude that the main stress relief mechanism for tantalum thin films is the beta to alpha phase transformation that occurs at 700 to 800°C.

A note on the computation of the incomplete beta function

Two efficient algorithms for computing the incomplete beta function are considered. They have comparable performance for a wide range of values of parameters and the variable.

繊維配向分布を考慮した繊維強化複合材料の剛性と変形の推定

This study was conducted to estimate the rigidity and the deformation of the fiber composites with complicated fiber orientation. The new method which was able to indicate various fiber oriented states by use of an incomplete beta function was proposed in the previous paper. In this paper, a generalized laminated theory was presented to analyse the stress-strain relation for the fiber composites with a certain fiber oriented distribution. The effect of fiber oriented distribution on the elastic moduli of these composites was discussed by the proposed formulation. The numerical results on the elastic moduli agreed well with the experimental data. Furthermore, the deformation behavior of a laminated plate was estimated by combining this lamination theory with the finite element method. The numerical example of the deformation under uniaxial loading was shown for a composite plate with a typical orientation state.

Some inequalities in elementary special functions with applications to nonparametric statistical inference

Some inequalities are established in P1(r, s) and P1(r+1, s), where P1(r, s) is the confidence coefficient of Wilks’ (1962) outer confidence interval (X(r) X(s)) for the quantile interval (ξp1, ξp2). An inequality concerning incomplete beta functions is also presented and it is shown to be an improved version of one of Koti's (1989) inequalities.

繊維強化複合材料における繊維配向分布の表示法

This paper deals with a method for representing the distribution of fiber orientation in fiber reinforced composites. It is well known that the mechanical properties of fiber composites depend on the elastic properties and volume fraction of the constituent materials, and are affected by the fabrication processing. In particular, the fiber orientation and content can be strongly influenced by the material flow in molding. Therefore, a knowledge of the fiber orientation distribution is a requirement for predicting the mechanical behavior of the fiber composites. In this paper, the fiber orientation density function using an incomplete beta function is proposed in order to express the distribution of fiber orientation for the fiber composite, and the mold flow of fibers in SMC laminates is discussed by the proposed formulation. It is clear that various distributions of fiber orientation can be expresssd by selecting proper paramater values.

The Cummulants and an Exact Expression of Generalized Logistic Distribution

A generalization of the logistic distribution proposed by Prentice (1975) is very useful for approximating the continuoui distributions and for data analysis. In this paper, an expression for the rth cumulant of the generalization is derived. The generalized logistic (GL) distribution of Prentice (1975) can be expressed as an incomplete beta function. In particular when both shape parameters are integers, the cumulative GL distribution function can be simply expressed as a sum of binomial probabilities. The exact expression of the cumulative GL distribution function when one of the shape parameters is not an integer is derived.

Some inequalities in incomplete beta functions by using statistical techniques

In this paper an inequality concerning incomplete beta functions is stated and proved. From this inequality several other inequalities are derived. The main inequality can be given by using the properties of incomplete beta functions but an interesting derivation by using statistical techniques is given.

Exact Simple Waves on Shear Flows in a Compressible Barotropic Medium

Self‐similar solutions describing simple waves on shear flows in a finite compressible barotropic atmosphere are found. These include the simple waves on shear flows in water as a special case. By making use of a number of transformations it becomes possible to write these solutions in an exact form. This form, though not explicit, is similar to the incomplete Beta function which seems to characterize this class of nonlinear physical problems.

Entropy loss and risk of improved estimators for the generalized variance and precision

Let the distributions of X(p×r) and S(p×p) be N(ζ, Σ⊗I r) and W p(n, Σ) respectively and let them be independent. The risk of the improved estimator for |Σ| or {ei329-1} based on X and S under entropy loss (=d/|Σ| −log(d/|Σ|)−1 or d|Σ|−log(d|Σ|)−1) is evaluated in terms of incomplete beta function of matrix argument and its derivative. Numerical comparison for the reduction of risk over the best affine equivariant estimator is given.

APL approximations for common statistical critical values

Evans and Gilfillan (1986) presented APL functions ZTAB, FTAB, CHlSQ, and BINOM, which produce quantile values from z scores, F values, x2 values, and frequencies, respectively. Quantiles of t scores may be obtained from function FTAB. Statistical analyses frequently requirethatcriticalvalues be obtained fromprobabilityvaluesunder the z, t, X , andF distributions when the requiredprobability valuesare not tabled. Oftensuch requirements arise under Bonferroni multiple comparison procedures, where in order to maintaina given level of familywise Type I error over many comparisons, the researcher mayneeda t or F criticalvaluecorresponding to a nontabled probability value. Other uses stem from the analysisof simpleeffects in factorial analysisof variance. This paper presents four APL functions that compute Z, t, X , and F critical values from upper-tailprobability values. The functions are ideal for constructing tables of critical values of the z, t, X , F, and incomplete beta distributions for given probability levels. The function that computes the F values (PTOF) depends on functions for the natural logarithmof the complete beta function (LNBETA), the incomplete beta function (BETAIN), and the inverse incomplete beta function (XINBTA). These three functions may be used for other applications. The functions are presented in Listing 1. Each of the seven functions has been written and tested in APL* PLUS/PCfromSTSC, Inc. mM APLdoesnotacceptthe diamond separator; enter the expressions separated by the diamonds in order from left to right. The execution times were obtained on an mM PC-AT compatible with an 80287 internal arithmetic coprocessor. Inverse Normal Function. Monadic function PTOZ acceptsa list of one or more upper-tailprobabilityvalues in its right argument and returns a vector of the corresponding approximate signed normal deviates. Thus, the result for rightargument [0.950.05 0.025] is [-1.645 1.645 1.960]. PTOZ uses the approximation given by Odeh and Evans (1974). Execution time for a random sampleof 100upper-tailprobability values was0.94 sec. Inverse t Function. Dyadic function PTOT accepts a list of one or more degrees-of-freedom values in its left argument and a list of associated upper-tail probability values in its right argument. PTOT returns a vector of the corresponding approximateStudent's t values. Thus,

FISHTAIL: PracticalF integration on small computers in BASIC and C

Computer algorithms can be used to calculate F sig­ nificance levels directly, thus avoiding the use of tables and resolving questions that arise when the degrees of freedom fall between the values given in an available pub­ lished table or when the exact probability level is of in­ terest. An algorithm is presented that finds F probabili­ ties and that represents a good compromise among the goals of accuracy, speed, and compactness. Unlike several others, this method has the capability of interpolation for approximations to significance levels, when noninteger values are given for the degrees of freedom. The fundamental idea underlying this algorithm was in­ troduced with the Biometrika tables ofthe incomplete beta function, from which the integrals ofF, t, and some other distributions can be obtained by the use of transforma­ tions (Kendall & Stuart, 1963, pp. 375-378). Following the suggestion of Bock (1975, p. 150), a computer al­ gorithm based on incomplete beta has been developed to bypass the tables. The method uses a finite number of terms of an infinite series that is known to converge to the desired value. The convergence is rapid enough for practical computations, under conditions that can be in­ duced. The method is based on the transformation x = V2/(V2 + vIF). In this expression, F is Fisher's F statistic with VI and V2 degrees of freedom for the numerator and the denomi­ nator, respectively. The transformed variate, x, is the var­ iate of a beta distribution with parameters a = v212 and b = vl 12 . The series expression for P, the tail area of Fisher's distribution beyond the point F (Bock, 1975, p. 150), is

An incomplete beta series with argument x(l-x)

An expression for calculating the incomplete beta cumulative distribution I x(a, b) is given in terms of the series and its derivative where For small values of w, the expression is convenient for numerical calculation. For integer values of a, b, closed form expressions are also given.

On the computation of F-cumulative probabilities

A recursive formula for evaluating incomplete beta integral is obtained by reducing both parameters simultaneously. Applying this identity, we derive an exact expression for F-cumulative probabilities. An algorithm is then derived by using this expression. Results from a single-precision FORTRAN program by using this algorithm are compared with that using MDFD subroutine of IMSL in both of accuracy and speed for degrees of freedom 1 lsequal; m, n $lsequal; 500. It is seen that both results coincide with at least 11 significant digits, and the new algorithm is consistently faster than MDFD by 10 to 15 percent.

On the t cumulative probabilities

In this paper we give an alternative to show the relationship between t and Beta using generalized logistic distribution. An expression for t-cumulative probabilities is then derived by evaluating the incomplete beta integral. An algorithm is also given for computing the t probabilities. This algorithm seems to have some desirable numerical properties.

A new approximation to the incomplete beta function

A new approximation to the incomplete beta function is proposed. This approximation compares favourably well with the Mudholkar-Chaubey (1976) approximation and it is infact superior for equal degrees of freedom. Moreover’ it is simpler to evaluate.

A new simple straightforward and practical asymptotic expansion for the incomplete beta function

It is shown that by using elementary transformations, the Incomplete Beta Function can be turned into a form suggesting that a close approximation by a normal integral can be obtained. By using standard methods this approximation is then developed in the form of an asymptotic series consisting of four exhibited correction terms to the main normal integral. An alternative means of deriving the third of these terms is shown confirming the correctness of the calculations (paradoxically the fourth correction term is easily derived and needs no such check). From this asymptotic series an asymptotic series for a corresponding standard normal variable is derived. The excellence of these approximations is exhibited by studying two limiting cases: that of equal indices (equivalent to a Student's distribution) and that of one index becoming infinite so that we have in place of an Incomplete Beta Function an Incomplete Gamma Function. Some similar recent results in the literature are commented on. Tables to show how excellent the expansion is, even down to parameter values as low as unity, are provided for a selection of parameter values and percentage points.

An approximation of F distribution by binomial probabilities

It is well known that the F distribution function can be expressed as an incomplete beta function. In particular when both degrees of freedom are even, the F distribution function may be simply expressed as a sum of binomial probabilities. In this note we consider the F distribution with at least one odd degree of freedom. By relating the F distribution to a generalized logistic distribution we obtain an approximation of the F distribution in terms of binomial probabilities. Because it is very close to the exact at the tails, the approximation is particularly suitable for estimating P-values of F ratio tests. All the computations involved can be carried out easily with only a hand calculator.

A representation in beta series for the exact density of some random volume contents

Consider r independent and identically distributed random points in a unit n-ball of which p are in the interior and rp are on the surface. These r points, via their convex hull, generate an r-simplex. This article deals with the exact density of the r-content when the points are uniformly distributed. The exact density of the r-content is obtained for the general values of the parameters r, n and p. A representation of the density is given as a mixture of beta type-1 densities so that one can evaluate various types of probabilities by using incom-plete beta tables.

Invariance group properties and exact solutions of equations describing time-dependent free surface flows under gravity

Using the method of infinitesimal transformations, a 6-parameter family of exact solutions describing nonlinear sheared flows with a free surface are found. These solutions are a hybrid between the earlier self-propagating simple wave solutions of Freeman, and decaying solutions of Sachdev. Simple wave solutions are also derived via the method of infinitesimal transformations. Incomplete beta functions seem to characterize these (nonlinear) sheared flows in the absence of critical levels.

Incomplete Laplace Integrals: Uniform Asymptotic Expansion with Application to the Incomplete Beta Function

The incomplete Laplace integral \[ \frac{1}{{\Gamma (\lambda )}}\int_\alpha ^\infty {t^{\lambda - 1} e^{ - zt} f(t)dt} \] is considered for large values of z. Both $\lambda $ and $\alpha $ are uniformity parameters in $[0,\infty )$. The basic approximant is an incomplete gamma function, that is, the above integral with $f = 1$. Also, a loop integral in the complex plane is considered with the same asymptotic features. The asymptotic expansions are furnished with error bounds for the remainders in the expansions. The results of the paper combine four kinds of asymptotic problems considered earlier. An application is given for the incomplete beta function. The present investigations are a continuation of earlier works of the author for the above integral with $\alpha = 0$. The new results are significantly based on the previous case.