Efficient Full Higher-Order Unification

We developed a procedure to enumerate complete sets of higher-order unifiers based on work by Jensen and Pietrzykowski. Our procedure removes many redundant unifiers by carefully restricting the search space and tightly integrating decision procedures for fragments that admit a finite complete set of unifiers. We identify a new such fragment and describe a procedure for computing its unifiers. Our unification procedure, together with new higher-order term indexing data structures, is implemented in the Zipperposition theorem prover. Experimental evaluation shows a clear advantage over Jensen and Pietrzykowski's procedure.

SUSY-breaking scenarios with a mildly violated $$\varvec{R}$$ symmetry

New realizations of the gravity-mediated SUSY breaking are presented consistently with an R symmetry. We employ monomial superpotential terms for the hidden-sector (goldstino) superfield and Kähler potentials parameterizing compact or non-compact Kähler manifolds. Their scalar curvature may be systematically related to the R charge of the goldstino so that Minkowski solutions without fine tuning are achieved. A mild violation of the R symmetry by a higher order term in the Kähler potentials allows for phenomenologically acceptable masses for the R axion. In all cases, non-vanishing soft SUSY-breaking parameters are obtained and a solution to the $$\mu $$ μ problem of MSSM may be accommodated by conveniently applying the Giudice–Masiero mechanism.

Global injectivity in second-gradient nonlinear elasticity and its approximation with penalty terms

We present a new penalty term approximating the Ciarlet–Nečas condition (global invertibility of deformations) as a soft constraint for hyperelastic materials. For non-simple materials including a suitable higher-order term in the elastic energy, we prove that the penalized functionals converge to the original functional subject to the Ciarlet–Nečas condition. Moreover, the penalization can be chosen in such a way that for all low-energy deformations, self-interpenetration is avoided completely already at all sufficiently small finite values of the penalization parameter. We also present numerical experiments in two dimensions illustrating our theoretical results and provide own MATLAB code available for download and testing.

Research on a Novel Compensation Algorithm for MEMS Sub-Pixel Displacement Measurement

The sub-pixel allocation is a key technology for achieving high-precision measurement for micro electro mechanical system (MEMS). In this paper, a novel sub-pixel compensation algorithm based on an improved gradient algorithm utilized in a rough sub-pixel position is proposed to compensate the insufficient accuracy extracted by the surface fitting. And compared with traditional gradient used in the integer pixel, the proposed algorithm can reduce the error introduced by abandoning the higher order term without using iteration and second-order Taylor formula. The experimental results show that the displacement parameters calculated by the proposed algorithm is more accurate, and the method has a good noise resistance, it can meet high-precision positioning of MEMS motion image.

Accurate Description of Near-Crack-Tip Fields for the Estimation of Inelastic Zone Extent in Quasi-Brittle Materials

A description of stress and displacement fields by means of the Williams power series using also higher-order terms is the focus of this paper. Coefficients of this series are determined via the over-deterministic method from the results of conventional finite element (FE) analysis. A study is conducted into the selection of the FE node set whose results are processed in this regression technique. Coefficients up to the twelfth term were determined with high precision. The effect of the position of the FE node set on the accuracy of the values of the higher-order term coefficients is reported.

Prototype testing of a composite-type active grating bender with Si substrate for surface profile high-order term elimination

The previous monolithic active grating bender design met some basic design requirements. However, after a real grating (BM-AGM) had been fabricated and installed for testing, the results showed that the usable length is a mere 60 mm because of the higher-order term error in the surface profile. A method was thus derived to eliminate the higher-order term error by modifying the width of the bender substrate through finite-element method simulation, reducing the residual error from about 100 nm to 6 nm. Owing to the closure of the grating department of Zeiss, ruling the monolithic bender is no longer available and the design has to be modified to a composite-type bender with Si substrate. A prototype was fabricated and assembled to examine all the design situations. The surface roughness of the width-modified Si substrate is around 30 nm before assembly. The residual error after assembly and bending is less than 10 nm. It proves that the design is feasible. However, due to the manufacturing capacity of the vendor, a short-length substrate is required and the design has to be modified. The detailed design modification and testing results are presented in this paper.