Bounds on Texture Coefficients

2003 ◽  
Vol 70 (2) ◽  
pp. 200-203 ◽  
Author(s):  
J. C. Nadeau ◽  
M. Ferrari
Keyword(s):  
Distribution Function ◽  
Orientation Distribution Function ◽  
Orientation Distribution ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Velocity Measurements ◽  
Texture Coefficients ◽  
Generalized Spherical Harmonics ◽  
Necessary And Sufficient ◽  
Microstructural Optimization

The orientation distribution function (ODF) is expanded in terms of generalized spherical harmonics and bounds on the resulting texture coefficients are derived. A necessary and sufficient condition for satisfaction of the normalization property of the ODF is also provided. These results are of significance in, for example, microstructural optimization of materials and predicting texture coefficients based on wave velocity measurements.

On the convergence of the supercritical branching processes with immigration

1977 ◽  
Vol 14 (2) ◽  
pp. 387-390 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Limit Distribution Function ◽  
Branching Process With Immigration ◽  
Necessary And Sufficient ◽  
Supercritical Branching Processes

It is shown for a supercritical branching process with immigration that if the log moment of the immigration distribution is infinite, then no sequence of positive constants {cn} exists such that {Xn/cn} converges in law to a proper limit distribution function F, except for the case F(0 +) = 1. Seneta's result [1] combined with the above-mentioned one imply that if 1 < m < ∞ then the finiteness of the log moment of the immigration distribution is a necessary and sufficient condition for the existence of some constants {cn} such that {Xn/cn} converges in law to a proper limit distribution function F, with F(0 +) < 1.

A New Method for Orientation Distribution Function Analysis

1985 ◽  
Vol 29 ◽  
pp. 443-449
Author(s):  
Munetsugu Matsuo ◽  
Koichi Kawasaki ◽  
Tetsuya Sugai
Keyword(s):  
Distribution Function ◽  
Spherical Harmonics ◽  
Orientation Distribution Function ◽  
Function Analysis ◽  
Orientation Distribution ◽  
Symmetry Operation ◽  
Composite Function ◽  
Odd Order ◽  
Pole Figures ◽  
Generalized Spherical Harmonics

AbstractAs a means for quantitative texture analysis, the crystallite orientation distribution function analysis has an important drawback: to bring ghosts as a consequence of the presence of a non-trivial kernel which consists of the spherical harmonics of odd order terms. In the spherical hamonic analysis, ghosts occur in the particular orientations by symmetry operation from the real orientation in accordance with the symmetry of the harmonics of even orders. For recovery of the odd order harmonics, the 9th-order generalized spherical harmonics are linearly combined and added to the orientation distribution function reconstructed from pole figures to a composite function. The coefficients of the linear combination are optimized to minimize the sum of negative values in the composite function. Reproducibility was simulated by using artificial pole figures of single or multiple component textures. Elimination of the ghosts is accompanied by increase in the height of real peak in the composite function of a single preferred orientation. Relative fractions of both major and minor textural components are reproduced with satisfactory fidelity In the simulation for analysis of multi-component textures.

Subexponential distribution functions

1980 ◽  
Vol 29 (3) ◽  
pp. 337-347 ◽  
Author(s):  
E. J. G. Pitman
Keyword(s):  
Distribution Function ◽  
Comparison Theorem ◽  
Distribution Functions ◽  
Sufficient Condition ◽  
Important Class ◽  
Necessary And Sufficient Condition ◽  
Closure Properties ◽  
Subexponential Class ◽  
Necessary And Sufficient ◽  
60 J 80

AbstractA distribution function (F on [0,∞) belongs to the subexponential class if and only if 1−F(2) (x) ~ 2(1−F(x)), as x→ ∞. For an important class of distribution functions, a simple, necessary and sufficient condition for membership of is given. A comparison theorem for membership of and also some closure properties of are obtained.1980 Mathematics subject classification (Amer. Math. Soe.): primary 60 E 05; secondary 60 J 80.

Acoustoelastic Response of a Polycrystalline Aggregate With Orthotropic Texture

1985 ◽  
Vol 52 (3) ◽  
pp. 659-663 ◽  
Author(s):  
G. C. Johnson
Keyword(s):  
Distribution Function ◽  
Orientation Distribution Function ◽  
Effective Properties ◽  
Orientation Distribution ◽  
Polycrystalline Aggregate ◽  
Third Order ◽  
Effective Response ◽  
Third Order Elastic Constants ◽  
Generalized Spherical Harmonics ◽  
Crystalline Symmetry

Estimates of the effective second-order and third-order elastic constants of a polycrystalline aggregate exhibiting texture are presented for the case of orthotropic sample symmetry and cubic crystalline symmetry. The nature of the texture is brought into the analysis through the crystallites’ orientation distribution function which is written in a series of generalized spherical harmonics. The effective response is evaluated using a Voigt-type procedure in which the crystal stiffnesses are averaged over the orientation distribution function. In evaluating the results, it is found that only seven terms in the expansion for the orientation distribution function are required for the exact representation of the effective properties.

Polycrystal elastic constants for triclinic crystal and physical symmetry

2006 ◽  
Vol 39 (4) ◽  
pp. 502-508 ◽  
Author(s):  
Peter R. Morris
Keyword(s):  
Distribution Function ◽  
Orientation Distribution Function ◽  
Orientation Distribution ◽  
Weight Functions ◽  
Higher Symmetry ◽  
X Ray ◽  
Pole Figures ◽  
Generalized Spherical Harmonics ◽  
Physical Symmetry ◽  
Symmetry Operations

The problem of obtaining the Voigt average for the elastic stiffnesses with texture-describing weight functions has been solved for triclinic crystal and physical symmetries. The average is obtained by expanding theTijklmnpq, which relate the elastic stiffnesses in the rotated reference frame, c^{\,\prime}_{ijkl}, to those of the principal elastic stiffnesses,cmnpq, in generalized spherical harmonics, multiplying by the orientation distribution function and integrating over all orientations. The condition imposed to assure a unique expansion results in the absence of terms with oddL, so that the results are completely determinable from conventional X-ray pole figures. This is the most general case, from which all higher-symmetry solutions may be obtained by application of symmetry operations. The Reuss average for elastic compliances may be obtained in a similar fashion.

Dispersive ordering by dilation

1990 ◽  
Vol 27 (02) ◽  
pp. 440-444 ◽  
Author(s):  
J. Muñoz-Perez ◽  
A. Sanchez-Gomez
Keyword(s):  
Distribution Function ◽  
Convex Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Dispersive Ordering ◽  
Necessary And Sufficient

In this paper a necessary and sufficient condition for the dispersive ordering in dilation sense is given by a convex function which is called the dispersive function and characterizes the distribution function. Some interesting properties of the ordering follow from this result.

Bi-Positive Sequences the Bilateral Moment Problem

1986 ◽  
Vol 29 (4) ◽  
pp. 456-462 ◽  
Author(s):  
Jaime Vinuesa ◽  
Rafael Guadalupe
Keyword(s):  
Distribution Function ◽  
Moment Problem ◽  
General Setting ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

AbstractWe pose a “moment problem” in a more general setting than the classical one. Then we find a necessary and sufficient condition for a sequence to have a solution of the “problem“where σ is a “distribution function”.

scholarly journals Tensorial Representation of the Orientation Distribution Function in Cubic Polycrystals

1992 ◽  
Vol 19 (3) ◽  
pp. 147-167 ◽  
Author(s):  
Maurizio Guidi ◽  
Brent L. Adams ◽  
E. Turan Onat
Keyword(s):  
Distribution Function ◽  
Spherical Harmonics ◽  
Orientation Distribution Function ◽  
Orientation Distribution ◽  
Basis Functions ◽  
Coordinate Frame ◽  
Crystallite Orientation ◽  
Fourier Representation ◽  
Irreducible Tensors ◽  
Generalized Spherical Harmonics

A precise definition for the crystallite orientation distribution function (codf) of cubic polycrystals is given in terms of the set of distinct orientations of a cube. Elements of the classical Fourier representation of the codf, in terms of (symmetrized) generalized spherical harmonics, are reviewed. An alternative Fourier representation is defined in which the coefficients of the series expansion are irreducible tensors. Since tensors can be defined without the benefit of a coordinate frame, the tensorial representation is coordinate free. A geometrical association between irreducible tensors and a bouquet of lines passing through a common origin is discussed. Algorithms are given for computing the irreducible tensors and basis functions for cubic polycrystals.

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