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We study sets of recurrence, in both measurable and topological settings, for actions of (ℕ, ×) and (ℚ>0, ×). In particular, we show that autocorrelation sequences of positive functions arising from multiplicative systems have positive additive averages. We also give criteria for when sets of the form {(an+b)1/(cn+d)ℓ: n ∈ ℕ} are sets of multiplicative recurrence, and consequently we recover two recent results in number theory regarding completely multiplicative functions and the Omega function. [ABSTRACT FROM AUTHOR]

We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if log r / log s is irrational and X and Y are × r - and × s -invariant subsets of [0, 1], respectively, then dim H (X + Y) = min (1 , dim H X + dim H Y) . Our main result yields information on the size of the sumset λ X + η Y uniformly across a compact set of parameters at fixed scales. The proof is combinatorial and avoids the machinery of local entropy averages and CP-processes, relying instead on a quantitative, discrete Marstrand projection theorem and a subtree regularity theorem that may be of independent interest. [ABSTRACT FROM AUTHOR]

Microalgae are a non-food grade, fast-growing, and non-land competitive biomass with relatively high energy content. The high lipid and mineral contents of microalgae render it beneficial for the production of biofuels and value-added products. The complete energy-effective harnessing of microalgae potentials concerning biodiesel production is faced with drawbacks because of the energy-intensive steps required to harvest and dry microalgae. The limitations have impelled the search for alternative and low-cost utilization of microalgae in wet form for biofuel production. Anaerobic digestion (AD) is among the wet techniques for the valorization of microalgae that is gaining immense research attention because of its simplicity. Also, the products can be recycled to reduce material costs. This review is focused on the recent trends and comparison of the AD process to maximize energy recovery from microalgae biomass and co-digestion of microalgae waste coupled to biodiesel production (after lipid extraction), respectively. The yield of methane gas in these two processes is compared and the pros and cons of biogas production using microalgae biomass and microalgae waste considering that the former produces biofuels and the later value-added products are discussed. [ABSTRACT FROM AUTHOR]

The optimization of a magnetohydrodynamic flow of Al2O3-water nanofluid was carried out numerically considering the combined effects of various parameters such as porous medium permeability, Forchheimer drag, slip length, conduction-radiation parameter and nanoparticle volume fraction on heat transfer and entropy generation. The numerical computations were made by using the shooting technique together with Runge-Kutta-Fehlberg method, and the results were compared with already published studies obtaining a very good agreement. Here, optimal operating conditions with minimum entropy and maximum or minimum heat transfer not yet reported in previous similar configurations were reached and the effects of porous medium in the presence of combined convective-radiative boundary conditions and nonlinear radiation heat flux were analyzed. Results showed that the global entropy increased with the porous medium permeability, while it decreased with the inertia parameter. In addition, optimum values of slip length and nanoparticle volume fraction that minimize the global entropy, were found. These optimum values of both quantities moved to higher values as the porous medium permeability increased. The Nusselt number was also explored for different conditions. Optimum values of Grashof number and conduction-radiation parameter with maximum heat transfer, as well as slip length with minimum heat transfer were found. These optimum values of Grashof moved to lower values as the permeability increased, while their optimum values shifted toward higher values with the inertia parameter. The obtained optimum values of slip length with minimum heat transfer moved to higher values when the permeability increased. Finally, four different models were defined to show the effects of uncertainties in thermophysical properties of nanofluid on heat transfer and entropy generation. These models were obtained from the combination of two relations used for both the dynamic viscosity and the thermal conductivity of nanofluid. It was seen that, independent of the model, optimal operating conditions were achieved for the explored values. [ABSTRACT FROM AUTHOR]

The restricted isometry property (RIP) is an important matrix condition in compressed sensing, but the best matrix constructions to date use randomness. This paper leverages pseudorandom properties of the Legendre symbol to reduce the number of random bits in an RIP matrix with Bernoulli entries. In this regard, the Legendre symbol is not special-our main result naturally generalizes to any small-bias sample space. We also conjecture that no random bits are necessary for our Legendre symbol-based construction. [ABSTRACT FROM AUTHOR]