Tungsten (W) films have many applications in the semiconducting industry for sensor technology. Deposition conditions can significantly impact the resulting W films in terms of the phases present (α-BCC or β-A12), microstructural grain orientation (texture), and residual strain. Tilt-A-Whirl methodology has been employed for the evaluation of a W film showing both texture and residual strain. Sin2(ψ) analysis of the film was performed to quantify the strongly tensile in-plane strain (+0.476%) with an estimated in-plane tensile stress of ~1.9 GPa. The 3D dataset was also evaluated qualitatively via 3D visualization. Visualization of 3D texture/strain data poses challenges due to peak broadening resulting from defocusing of the beam at high ψ tilt angles. To address this issue, principal component analysis (PCA) was employed to diagnose, model, and remove the broadening component from the diffraction data. Evaluation of the raw data and subsequent corrected data (after removal of defocusing effects) has been performed through projection of the data into a virtual 3D environment (via CAD2VR software) to qualitatively detect the impact of residual strain on the observed pole figure.

One of the best-known methods for discriminating λ-terms with respect to β-convertibility is due to Corrado Böhm. The idea is to compute the infinitary normal form of a λ-term M, the Böhm Tree (BT) of M. If λ-terms M, N have distinct BTs, then M ≠βN, that is, M and N are not β-convertible. But what if their BTs coincide? For example, all fixed point combinators (FPCs) have the same BT, namely λx.x(x(x(. . .))).

We introduce a clocked λ-calculus, an extension of the classical λ-calculus with a unary symbol τ used to witness the β-steps needed in the normalization to the BT. This extension is infinitary strongly normalizing, infinitary confluent and the unique infinitary normal forms constitute enriched BTs, which we call clocked BTs. These are suitable for discriminating a rich class of λ-terms having the same BTs, including the well-known sequence of Böhm's FPCs.

We further increase the discrimination power in two directions. First, by a refinement of the calculus: the atomic clocked λ-calculus, where we employ symbols τp that also witness the (relative) positions p of the β-steps. Second, by employing a localized version of the (atomic) clocked BTs that has even more discriminating power.

We work in , the untyped λ-calculus in which all unsolvables are identified. We resolve a conjecture of Barendregt asserting that the range of a definable map is either infinite or a singleton. This is refuted by constructing a λ-term Ξ such that ΞM = ΞI ⇔ ΞM ≠ ΞΩ. The construction generalizes to ranges of any finite size, and to some other sensible lambda theories.