We describe the design and fabrication of miniaturized origami structures based on thin-film shape memory alloys. These devices are attractive for medical implants, as they overcome the opposing requirements of crimping the implant for insertion into an artery while keeping sensitive parts of the implant nearly stress-free. The designs are based on a group theory approach in which compatibility at a few creases implies the foldability of the whole structure. Importantly, this approach is versatile and thus provides a pathway for patient-specific treatment of brain aneurysms of differing shapes and sizes. The wafer-based monolithic fabrication method demonstrated here, which comprises thin-film deposition, lithography, and etching using sacrificial layers, is a prerequisite for any integrated self-folding mechanism or sensors and will revolutionize the availability of miniaturized implants, allowing for new and safer medical treatments.

No abstract

In this paper we prove that, after an appropriate rescaling, the sum of moments E N ( s ) Tr | H | 2 k + 2 + | H | 2 k of an N × N Hermitian matrix H sampled according to the generalized Cauchy (also known as Hua–Pickrell) ensemble with parameter s > 0 is a continuous-Hahn polynomial in the variable k. This completes the picture of the investigation that began in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) where analogous results were obtained for the other three classical ensembles of random matrices, the Gaussian, the Laguerre and Jacobi. Our strategy of proof is somewhat different from the one in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) due to the fact that the generalized Cauchy is the only classical ensemble which has a finite number of integer moments. Our arguments also apply, with straightforward modifications, to the other three cases studied in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) as well. We finally obtain a differential equation for the one-point density function of the eigenvalue distribution of this ensemble and establish the large N asymptotics of the moments.

The problem of convergence of the joint moments, which depend on two parameters s and h, of the characteristic polynomial of a random Haar-distributed unitary matrix and its derivative, as the matrix size goes to infinity, has been studied for two decades, beginning with the thesis of Hughes (On the characteristic polynomial of a random unitary matrix and the Riemann zeta function, PhD Thesis, University of Bristol, 2001). Recently, Forrester (Joint moments of a characteristic polynomial and its derivative for the circular $$\beta $$ β -ensemble, arXiv:2012.08618, 2020) considered the analogous problem for the Circular $$\beta $$ β -Ensemble (C$$\beta $$ β E) characteristic polynomial, proved convergence and obtained an explicit combinatorial formula for the limit for integer s and complex h. In this paper we consider this problem for a generalisation of the C$$\beta $$ β E, the Circular Jacobi $$\beta $$ β -ensemble (CJ$$\beta \text {E}_\delta $$ β E δ ), depending on an additional complex parameter $$\delta $$ δ and we prove convergence of the joint moments for general positive real exponents s and h. We give a representation for the limit in terms of the moments of a family of real random variables of independent interest. This is done by making use of some general results on consistent probability measures on interlacing arrays. Using these techniques, we also extend Forrester’s explicit formula to the case of real s and $$\delta $$ δ and integer h. Finally, we prove an analogous result for the moments of the logarithmic derivative of the characteristic polynomial of the Laguerre $$\beta $$ β -ensemble.