The problem of maximizing the L p norms of chords connecting points on a closed curve separated by arc length u arises in electrostatic and quantum-mechanical problems. It is known that among all closed curves of fixed length, the unique maximizing shape is the circle for 1 p 2, but this is not the case for sufficiently large values of p. Here we determine the critical value p c (u) of p above which the circle is not a local maximizer finding, in particular, that p c (1 2 L)= 5 2. This corrects a claim made in [P. Exner, EM Harrell, M. Loss, Lett. Math. Phys. 75 (2006) 225].

Non-line-of-sight (NLOS) imaging enables people to see a hidden scene based on multiple interaction information between the object and the carrier. There have been numerous studies focusing on the physical modeling of photon scattering, but few have explored the detection process, which also plays a vital role. In this paper, we put forward a novel, to the best of our knowledge, detection methodology for NLOS imaging based on time-sequential first photon (TSFP) data. We verify the method with both synthetic and experimental data, showing a dramatic reduction in acquisition time cost compared with traditional methods for the same reconstruction quality. This work may contribute to real-time and photon-starved NLOS imaging for practical applications.

We present a generalized learning with errors (LWE) problem, which is essentially a simple and direct extension of the original LWE problem to the case of matrices. Then we use this new version of LWE problem, which we call matrix LWE problem to build new cryptographic schemes, which include a new scalable key distribution scheme, a new key exchanges scheme and a new simple identity-based encryption scheme.

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