The Geometry of Enhancement in Multiple Regression

In linear multiple regression, “enhancement” is said to occur when R ²=b′r>r′r, where b is a p×1 vector of standardized regression coefficients and r is a p×1 vector of correlations between a criterion y and a set of standardized regressors, x. When p=1 then b≡r and enhancement cannot occur. When p=2, for all full-rank R _xx≠I, R _xx=E[xx′]=V Λ V′ (where V Λ V′ denotes the eigen decomposition of R _xx; λ ₁>λ ₂), the set B₁:={b_i:R²=b_i￠r_i=r_i￠r_i;0 < R² ￡ 1}\boldsymbol{B}_{1}:=\{\boldsymbol{b}_{i}:R^{2}=\boldsymbol{b}_{i}'\boldsymbol{r}_{i}=\boldsymbol{r}_{i}'\boldsymbol{r}_{i};0R² ￡ 1;R²l_p ￡ r_i￠r_i < R²}0p≥3 (and λ ₁>λ ₂>⋯>λ _p ), both sets contain an uncountably infinite number of vectors. Geometrical arguments demonstrate that B ₁ occurs at the intersection of two hyper-ellipsoids in ℝ ^p . Equations are provided for populating the sets B ₁ and B ₂ and for demonstrating that maximum enhancement occurs when b is collinear with the eigenvector that is associated with λ _p (the smallest eigenvalue of the predictor correlation matrix). These equations are used to illustrate the logic and the underlying geometry of enhancement in population, multiple-regression models. R code for simulating population regression models that exhibit enhancement of any degree and any number of predictors is included in Appendices A and B.     

Deformation mechanism of long-period TiAl 2

The microstructure of long-period TiAl 2 deformed at room temperature has been studied by means of transmission electron microscopy. Dislocations, stacking faults and twins were found to contribute to the deformation. A screw superdislocation with Burgers vector d 110] dissociates into two ½ d 110] super-partial dislocations associated with an antiphase boundary. The ½; d 110] super-partial dislocation further dissociates into two Shockley partial dislocations associated with a portion of complex intrinsic stacking fault on the closest-packed {111} plane. The propagation of stacking faults on successive {111} planes yields an order twin.     

Dissociation of dislocations in MgAl2O4 spinel deformed at low temperatures

Abstract

When spinel is deformed in compression at 400°C along 〈110〉, the primary slip plane is found to be {111} with cross-slip occurring on a {001} plane. A comparison of weak-beam images of dislocations from both systems indicates that all dislocations which belong to the primary slip plane are dissociated out of the {111} plane independent of the character of the dislocation. It is proposed that deformation occurs by motion of dislocations in their dissociated state and that the partial dislocations actually glide on parallel glide planes. Movement of these dissociated dislocations is then accompanied by a concurrent migration of the stacking fault which takes place by a local shuffling of the cations. A stacking fault energy for conservative dissociation at 400°C on {001} of 530±90mJ m^?2 has been determined from weak-beam images of screw dislocations.     

Investigation of {111} stacking faults and nanotwins in epitaxial BaTiO3 thin films by high-resolution transmission electron microscopy

{111} stacking faults and nanotwins in epitaxial BaTiO₃ thin films on MgO substrates have been investigated by high-resolution transmission electron microscopy. In many cases, the stacking faults and nanotwins were found to be accompanied by partial dislocations. These partial dislocations can be classified as two different types, analogous to the situation in the fcc structure. One is of the Shockley type with the Burgers vector (a/3)<112>. The other is of the Frank type with the Burgers vector (a/3)<111>. The movements of both types of partial can lead to the {111} stacking faults and the {111} twins observed in these films.     

The order–dsorder transition at twin boundaries in Cu3Au as studied by TEM

Abstract

The order-disorder phase transition at ∑ = 3{111}- and {211}-type twin boundaries has been studied in the L1²-ordered alloy Cu³Au employing in situ heating in a transmission electron microscope. Evidence is presented for an order-disorder phase transition occurring in these boundaries prior to the bulk transition. The temperature difference ΔT between the transition temperature of both boundary types and the bulk is estimated as 0.5K <ΔT<2K. No difference in T _c for the twin boundaries can be established as yet. The nature of the order-disorder transition in both twin boundaries is presumably a second-order phase transition.     

Anisotropic BaTiO3 thin films grown on MgO-buffered R-plane sapphire

Epitaxial BaTiO₃ thin films grown on MgO-buffered R-plane-cut Al₂O₃ substrates show a non-uniform distribution of planar defects. Transmission electron microscopy allows the determination of the distributions of planar defects with respect to the Al₂O₃ substrate. The BaTiO₃ film, the MgO buffer and the Al₂O₃ substrate have an orientation relationship of [100]_BaTiO3//[100]_MgO//[1120]_Al2O3, (010)_BaTiO3//(010)_MgO//(1104)_Al2O3 and (001)_BaTiO3//(001)_MgO// ≈ (1102)_Al2O3 (about 5°; deviation). Under the above relationship, most of the {111} stacking faults occur on (111) and (111) planes, that is two of four sets of {111}_BaTiO3, whereas cracks in the BaTiO₃ film (and MgO buffer) are situated on the (100) planes. The anisotropic distribution of planar defects is regarded as a consequence of an anisotropic stress in the films generated during their fabrication.     

High-resolution characterization of the precipitation behavior of an Al–Zn–Mg–Cu alloy

The metastable particles in an Al–Zn–Mg–Cu alloy have been examined at atomic-resolution using high-angle annular dark field (HAADF) imaging. In under-aged conditions, thin η′ plates were formed with a thickness of seven atomic planes parallel to the {111}_Al planes. The five inner planes of the η′ phase appear to be alternatively enriched in Mg and Zn, with two outer planes forming distinct Zn-rich interfacial planes. Similar Zn-rich interfacial enrichment has also been identified for the η phase, which is a minimum 11-plane thick structure. In rare instances, particles less than seven planes were found indicating a very early preference for seven-layer particle formation. Throughout the aging, the plate thickness appears constant, while the plate radius increases and no particles between 7 and 11 planes were observed. Based on the HAADF contrast, our observations do not support the η′ models previously set forth by other authors. Clear structural similarities between η′ and η were observed, suggesting that drawing distinctions between η′ and η phases may not be necessary or useful.     

The crystallographic structure of λ transition-alumina compared with the periodic antiphase boundary structure of a Ga2O3-rich Mg gallate

One of the high-temperature metastable phases found in the MgGa₂O₄–Ga₂O₃ system is the ε_Mg non-stoichiometric Mg gallate. It has a one-dimensional periodic antiphase boundary (PAPB) structure based on the spinel structure. The APBs are parallel to the {3?1?0} planes and the APB vectors are of the 1/4?1 1 0? type when referred to the spinel structure. This generally leads to the existence of 12 monoclinic twin variants in the studied samples. By comparing the X-ray or electronic diffraction patterns of both phases, it will be shown that the recently discovered?λ?transition-alumina has the same type of structure.     

Cathodoluminescence-mode imaging of dislocations in zinc oxide

Abstract

The cathodoluminescence mode in the scanning electron microscope has been used to image bands of dislocations in zinc oxide. It has been shown that dislocations lying on pyramidal, {1012}, planes are generated beneath indentations on {0001} surfaces. Basal plane dislocations lying parallel to the surface are also observed. Indentations on {1100} prismatic surfaces result in slip on {0001} and {0110} planes.     

High-angle annular dark field scanning transmission electron microscopy of the antiphase boundary in a rapidly solidified B2 type TiPd compound

High-angle annular dark field scanning transmission electron microscopy (HAADF-STEM) with Z-contrast is applied to characterize the antiphase boundary (APB) of the B2 structure in a rapidly solidified TiPd melt-spun compound. The atomic shift associated with the R ?=?(1/2) a ₀ ?111? type displacement vector is directly observed at the boundary. A microstructure modification of the melt-spun compound with the cooling rate during solidification is also described.     

Non-dissociation of Lomer–Cottrell dislocations and 〈110〉{001} slip in silicon

Abstract

Two unexpected features have been revealed by transmission electron microscopy investigations of deformed silicon bicrystals: (a) slip of a/2 〈110〉 dislocations on {001} planes and (b) non-dissociation of Lomer–Cottrell dislocations formed by intersecting slip dislocations.     

Dislocation relaxation and alloy softening of bcc alloys

Recently, low-frequency internal friction measurements on a series of Fe–Cr alloys by Konstantinovi? and Terentyev [M.J. Konstantinovi? and D. Terentyev, Mater. Sci. Eng. A 521–522 (2009) p.106] have demonstrated that increasing Cr concentrations lead to an increase in the strength of the β-relaxation at the cost of the γ-relaxation (Chambers’ notation). In the same concentration and temperature regime, the alloys show alloy softening. It is argued that both phenomena are due to the same process, namely the influence of foreign atoms on the transformation of the cores of a ₀?1 1 1?/2 screw dislocations from their low-temperature configuration, capable of forming kink pairs on {1 1 0} planes, to their high-temperature configuration with kink-pair generation on {2 1 1} planes.     

On the asymmetrical dissociation of Lomer dislocations

Abstract

Using anisotropic elasticity theory, Korner, Schmid and Prinz have shown that Lomer-Cottrell dislocations in the f.c.c. structure have asymmetrical equilibrium positions. This is also true for the isotropic theory, but in addition it can also be demonstrated in a very simple way that the ratio of the dissociation widths on the two {111} planes is material independent. This result seems to be general even for the anisotropic case. However, careful observations of Lomer dislocations in copper by weak-beam transmission electron microscopy indicate clearly that they are not dissociated.     

Newly observed microscopic planar defects on {111} in natural diamond

Abstract

Circular disc-crack-like defects randomly distributed on {111} planes have been discovered by 300kV transmission electron microscopy in diamond that had crystallized on non-faceted growth surfaces of mean orientation {100}. In a sample of ～ 100 defects, disc diameters were narrowly distributed about a mean of 1·2 μm and one in seven discs were located on a coplanar stacking fault of average diameter 2 μm. Discs totally enclosed within the specimen exhibit concentric moiré fringes covering their image areas when viewed obliquely, showing them to be dilated into thin lenticles. Internal pressures, deduced from moiré-fringe counts on six lenticles, were in the range 1·5 to 1·8 GPa.     

Crystallographic orientation dependence of nanoindentation hardness in austenitic phase of stainless steel

ABSTRACT

This paper presents the indentation hardness evolution in different in-plane directions of austenite grains whose {111} planes are parallel to the sample surface determined by nanoindentation tests and electron back scattering diffraction (EBSD) analysis. A scanning electron microscopy (SEM) image of the indentation surface around one of the indents indicated the activation of two sets of slip planes with respect to each of the three indenter surfaces for a Berkovich tip. The identification of the slip traces by EBSD data analysis is in accord with Schmid`s law. We therefore proposed a new approach for defining the orientation parameter to interpret the indentation hardness. The orientation parameter was shown to be the minimum value of the three maximum Schmid factors on the secondarily activated slip planes in three directions for a Berkovich tip. Indentation hardness increased with the decrease in the orientation parameter and was dependent on in-plane orientation.     

Planar dissociations and recombination energy of [110] superdislocations in Ni3Al: Generalized Peierls model in combination with ab initioelectron theory

Planar dissociation configurations of the [110] superdislocation in stoichiometric Ni₃Al are investigated by the generalized Peierls model using the {111} γ-surface energies calculated by the ab initio electron theory. The dissociation into four Shockley partials with the formation of an antiphase boundary and two complex stacking fault (CSF) ribbons turned out to be the energetically most favourable configuration. For the edge superdislocation the obtained dissociation widths should be large enough to be resolved in a weakbeam image, whereas for the corresponding configuration of the screw superdislocation the CSF ribbon has a rather small width which may be beyond the limit of the resolution. A twofold dissociation involving a superlattice intrinsic stacking fault is energetically less favourable. The recombination energy of the two Shockley partials in one superpartial is found to be 0.24eV/b.     

HRTEM observation of interfacial dislocations at faceted Al–Pb interfaces

Al–Pb ribbons containing 1?at.%?Pb have been produced by melt-spinning and subsequently investigated by high-resolution transmission electron microscopy. It is shown that the lattice mismatch of about 22% between the nanometre-sized Pb inclusions and the surrounding Al matrix is accommodated by a periodic array of misfit dislocations at the Al–Pb interface. The closing failures of Burgers circuits drawn around misfit dislocations on {111} and {100} facets identify the corresponding Burgers vectors as (a ₀/4)? 211 ? and (a ₀/2)? 110?, respectively. The Burgers vector of (a ₀/4)? 211? corresponds to the projected edge part of a 60° (a ₀/2)? 110? dislocation. The Pb inclusions themselves appear to be free of defects.     

Types of \mathsf{I} -Free Hereditary Right Maximal Terms

The implicational fragment of the relevance logic “ticket entailment” is closely related to the so-called hereditary right maximal terms. I prove that the terms that need to be considered as inhabitants of the types which are theorems of T_→ are in normal form and built in all but one case from and only. As a tool in the proof ordered term rewriting systems are introduced. Based on the main theorem I define FIT_→ – a Fitch-style calculus (related to FT_→) for the implicational fragment of ticket entailment.     

Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic

A scaled difference test statistic [(T)\tilde]_d\tilde{T}{}_{d} that can be computed from standard software of structural equation models (SEM) by hand calculations was proposed in Satorra and Bentler (Psychometrika 66:507–514, 2001). The statistic [(T)\tilde]_d\tilde{T}_{d} is asymptotically equivalent to the scaled difference test statistic [`(T)]<sub>d</sub>\bar{T}_{d} introduced in Satorra (Innovations in Multivariate Statistical Analysis: A Festschrift for Heinz Neudecker, pp. 233–247, 2000), which requires more involved computations beyond standard output of SEM software. The test statistic [(T)\tilde]<sub>d</sub>\tilde{T}_{d} has been widely used in practice, but in some applications it is negative due to negativity of its associated scaling correction. Using the implicit function theorem, this note develops an improved scaling correction leading to a new scaled difference statistic [`(T)]_d\bar{T}_{d} that avoids negative chi-square values.     

A first-principles study of the formation of θ′′ phase in Al–Cu alloys

The θ′′-Al₃Cu phase plays an important role in the precipitation process of Al–Cu alloys. This phase has a sandwich structure—every two {200}_Cu layers are separated by three {200}_Al layers. To analyse the formation mechanism of this structure, the elastic strain energy of the {200}_Cu and {200}_Al layers, and the chemical bonding energy that reflects the interaction between the electrons in Cu and neighbouring Al atoms are calculated and analysed by first-principles calculations, projected density of states and Bader analysis. Our computation results reveal that this sandwich structure is energetically preferred in the competition of elastic strain and chemical bonding energies. To minimise the elastic strain energy of {200}_Al and {200}_Cu layers, the {200}_Cu layers prefer being apart from each other, whereas the chemical bonding energy favours the opposite arrangement because the intermetallic bond between Al and Cu atoms may form through p-d hybridization.