We present two case studies in formal reasoning about untyped λ-calculus in Coq, using both first-order and higher-order abstract syntax. In the first case, we prove the equivalence of three definitions of α-equivalence; in the second, we focus on properties of substitution. In both cases, we deal with contexts, which are rendered by means of higher-order terms (functions) in the metalanguage. ...

The sensitivities of current gravitational-wave detectors are limited around signal frequencies 100 Hz by mirror thermal noise. One proposed option to reduce this noise is operate the in a higher-order spatial laser mode. This operation would require high-power input beam such Here, we discuss generation Hermite–Gaussian modes HG2,2, HG3,3, and HG4,4 using one water-cooled light modulator (SLM)...

Consider the much-discussed case of the distracted driver, who is alleged to successfully navigate his car for miles despite being completely oblivious to his visual states. Perhaps he is deeply engrossed in the music playing over the radio or in philosophical reflection, and as a result he goes about unaware of the scene unfolding before him on the road. That the distracted driver has visual e...

Both syntax-phonology and syntax-semantics interfaces in Higher Order Grammar (HOG) are expressed as axiomatic theories in higher-order logic (HOL), i.e. a language is defined entirely in terms of provability in the single logical system. An important implication of this elegant architecture is that the meaning of a valid expression turns out to be represented not by a single, nor even by a few...

Based on the second-order moments definition, beam propagation factor of a new mathematical model Hollow higher-order Cosh-Gaussian (HhCG) beams is investigated. Analytical formula for M2-factor HhCG derived, which depends hollowness parameter l, order n, cosh part δ and waist ω0. The influences these parameters are illustrated with numerical examples. result provides more general characteristi...