Air Force Institute of Technology

Let m≤n and r≥1. We study one-parameter deformations F:[0,1]×(Rn,0)→(Rm,0) of class C1,r+1 relative to a fixed closed germ Z⊂(Rn,0) such that SingFt⊂Z for all t∈[0,1]. Writing P=∂tF and denoting by ν(dxF) the minimal singular value of the spatial differential, we prove that if|∂xsP(t,x)|≤Cν(dxF(t,x))r+1−|s|dist(x,Z)1+α,|s|≤r, for some α>0, then F is Cr-trivial near Z along [0,1]; in fact, dist(x,Z)1+α may be replaced by dist(x,Z)θ with any θ>0, so the borderline case α=0 is included. No regularity of Z is assumed. The trivializing isotopy is the flow of the Thom–Levine field W=−RP (R being the Moore–Penrose right inverse of dxF), extended by zero across Z; it fixes Z, has the r-jet of the identity along Z and is bi-Lipschitz with explicit distortion bounds. We also give a minor-based reformulation of the hypothesis and a pathwise multiparameter variant.

Experimental research was conducted on the fatigue life of Selective Laser Melted Ti6Al-4v. A thorough understanding of the fatigue life performance for additively manufactured parts is necessary before such parts are utilized as production end-items for real-world applications such as the rapid, on-demand, 3D-printing of aircraft replacement parts. This research experimentally examines the fatigue life of Ti-6Al-4v material specimens built directly to net shape and then either stress-relieved or Hot Isostatically Pressed (HIP). Experimental results will help determine whether HIP effectively reduces porosity and increases fatigue life when the specimen surface is not machined to remove surface roughness from the additive manufacturing process.

Although additive manufacturing presents a novel and customizable approach to fabricating tungsten (W) alloys, the layer-by-layer production technique presents a unique set of processing challenges which are further exacerbated by the metal’s refractory properties. In this study, additions of niobium (Nb) were found to improve the mechanical properties and oxidation resistance of additively manufactured W. Microstructural investigation revealed the effect of Nb on the cracking, grain size, and texture in the as-built material. Mechanical compression testing showed significant improvement in ductility with increasing Nb content. The low concentrations of Nb used did not induce large changes in the yield strength of the material. The work shows that Nb may be a useful alloying element for improving additive manufacturing of W.

We show that there does not exist a complex d× n equiangular tight frame with d^2-d+1<n<d^2. The proof, which originated from an internal model at OpenAI, mimics the relationship between real equiangular tight frames and strongly regular graphs.

We build on the recent breakthrough of Faulhuber, Petersen, van Velthoven, and Voigtlaender that disproved the HRT conjecture with a Schwartz function and a 12-point configuration. We give a human-readable treatment of their mechanism and find HRT counterexample functions with exponential (or faster) decay. By a result of Bownik and Speegle, this is the fastest possible decay for an HRT counterexample, up to a logarithmic factor in the exponent.