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Acoustic detection has many applications across science and technology, from medical to imaging and communications. However, most acoustic sensors have a common limitation in that the detection must be near the acoustic source. Alternatively laser interferometry with picometer-scale motional displacement detection can rapidly and precisely measure sound induced minute vibrations on remote surfaces. Here we demonstrate the feasibility of sound detection up to 100 kHz at remote sites with ~ 60 m of optical path length via laser homodyne interferometry. Based on our ultrastable Hz-linewidth laser with 10-15 fractional stability, our laser interferometer achieves 0.5 pm/Hz1/2 displacement sensitivity near 10 kHz, bounded only by laser frequency noise over 10 kHz. Between 140 Hz to 15 kHz, we achieve a homodyne acoustic sensing sensitivity of sub-nm/Pa across our conversational frequency overtones. The minimal sound pressure detectable over 60 m of optical path length is ~ 2 mPa, with dynamic ranges over 100 dB. With the demonstrated standoff picometric distance metrology, we successfully detected and reconstructed musical scores of normal conversational volumes with high fidelity. The acoustic detection via this precision laser interferometer could be applied to selective area sound sensing for remote acoustic metrology, optomechanical vibrational motion sensing and ultrasensitive optical microphones at the laser frequency noise limits., Comment: 25 pages, 10 figures

We consider primitive substitution tilings on R^d whose expansion maps are unimodular. We assume that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, we can construct a cut-and-project scheme with a Euclidean internal space. Under some additional condition, we show that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.

For the noncrystallographic Coxeter groups of type $H$, we construct their Gr\"obner-Shirshov bases and the corresponding standard monomials., Comment: 14 pages, 2018 KMS Spring annual meeting

We construct a Gr\"obner-Shirshov basis of the Temperley-Lieb algebra $\mathfrak{T}(d,n)$ of the complex reflection group $G(d,1,n)$, inducing the standard monomials expressed by the generators $\{E_i\}$ of $\mathfrak{T}(d,n)$. This result generalizes the one for the Coxeter group of type $B_n$ in \cite{KimSSLeeDI}. We also give a combinatorial interpretation of the standard monomials of $\mathfrak{T}(d,n)$, relating to the fully commutative elements of the complex reflection group $G(d,1,n)$. In this way, we obtain the dimension formula of $\mathfrak{T}(d,n)$., Comment: 23 pages. arXiv admin note: text overlap with arXiv:1808.05026

For Temperley-Lieb algebras of types $B$ and $D$, we construct their Gr\"obner-Shirshov bases and the corresponding standard monomials., Comment: 9 pages. Submitted to a Journal on April 2017