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The Frobenius-Perron theory of an endofunctor of a k \Bbbk -linear category (recently introduced in Chen et al. [Algebra Number Theory 13 (2019), pp. 2005–2055]) provides new invariants for abelian and triangulated categories. Here we study Frobenius-Perron type invariants for derived categories of commutative and noncommutative projective schemes. In particular, we calculate the Frobenius-Perron dimension for domestic and tubular weighted projective lines, define Frobenius-Perron generalizations of Calabi-Yau and Kodaira dimensions, and provide examples. We apply this theory to the derived categories associated to certain Artin-Schelter regular and finite-dimensional algebras.

We propose SenseHash, a novel design for the lightweight in-hardware mystification of the sensed data at the origin. The framework aims to ensure the privacy of sensitive sensor values while preserving their utility. The sensors are assumed to interface to various (potentially malicious) communication and computing components in the Internet-of-things (IoT) and other emerging pervasive computing scenarios. The primary security primitives of our work are Locality Sensitive Hashing (LSH) combined with Differential Privacy (DP) and secure construction of LSH. Our construction allows (i) sub-linear search in sensor readings while ensuring their security against triangulation attack, and (ii) differentially private statistics of the readings. SenseHash includes hardware architecture as well as accompanying protocols to efficiently utilize the secure readings in practical scenarios. Alongside these scenarios, we present an automated workflow to generalize the application of the mystified readings. Proof-of-concept FPGA implementation of the system demonstrates its practicability and low overhead in terms of hardware resources, energy consumption, and protocol execution time.

We report the observation of terahertz field-induced second harmonic (TFISH) generation produced by directly mixing an optical probe beam onto femtosecond plasma filaments. The produced TFISH signal is spatially separated from the laser-induced supercontinuum by impinging on the plasma at a noncollinear angle. The conversion efficiency of the fundamental probe beam to its second harmonic (SH) beam is greater than 0.02%, which represents a record in optical probe to TFISH conversion efficiency that is nearly five orders of magnitude larger than previous experiments. We also present the terahertz (THz) spectral buildup of the source along the plasma filament and retrieve coherent terahertz signal measurements. This method of analysis has the potential to provide local electric field strength measurements inside of the filament.

We have derived a 3-D kinetic-based discrete dynamic system (DDS) from the lattice Boltzmann equation (LBE) for incompressible flows through a Galerkin procedure. Expressed by a poor-man lattice Boltzmann equation (PMLBE), it involves five bifurcation parameters including relaxation time from the LBE, splitting factor of large and sub-grid motion scales, and wavevector components from the Fourier space. Numerical experiments have shown that the DDS can capture laminar behaviors of periodic, subharmonic, n-period, and quasi-periodic and turbulent behaviors of noisy periodic with harmonic, noisy subharmonic, noisy quasi-periodic, and broadband power spectra. In this work, we investigated the effects of bifurcation parameters on the capturing of the laminar and turbulent flows in terms of the convergence of time series and the pattern of power spectra. We have found that the 2nd order and 3rd order PMLBEs are both able to capture laminar and turbulent flow behaviors but the 2nd order DDS performs better with lower computation cost and more flow behaviors captured. With the specified ranges of the bifurcation parameters, we have identified two optimal bifurcation parameter sets for laminar and turbulent behaviors. Beyond this work, we are exploring the regime maps for a deeper understanding of the contributions of the bifurcationmore »parameters to the capturing of laminar and turbulent behaviors. Surrogate models (to replace the PMLBE) are being developed using deep learning techniques to overcome the overwhelming computation cost for the regime maps. Meanwhile, the DDS is being employed in the large eddy simulation of turbulent pulsatile flows to provide dynamic sub-grid scale information.« less