Search for: All records

Abstract We describe an algorithm for computing, for all primes$$p \le X$$ $p\le X$, the trace of Frobenius atpof a hypergeometric motive over$$\mathbb {Q}$$ $Q$in time quasilinear inX. This involves computing the trace modulo$$p^e$$ $p^e$for suitablee; as in our previous work treating the case$$e=1$$ $e=1$, we combine the Beukers–Cohen–Mellit trace formula with average polynomial time techniques of Harvey and Harvey–Sutherland. The key new ingredient for$$e>1$$ $e>1$is an expanded version of Harvey’s “generic prime” construction, making it possible to incorporate certainp-adic transcendental functions into the computation; one of these is thep-adic Gamma function, whose average polynomial time computation is an intermediate step which may be of independent interest. We also provide an implementation in Sage and discuss the remaining computational issues around tabulating hypergeometricL-series. 

Abstract The next radical change in the thermal management of data centers is to shift from conventional cooling methods like air-cooling to direct liquid cooling to enable high thermal mass and corresponding superior cooling. There has been in the past few years a limited adoption of direct liquid cooling in data centers because of its simplicity and high heat dissipation capacity. Single-phase engineered fluid immersion cooling has several other benefits like better server performance, even temperature profile, and higher rack densities and the ability to cool all components in a server without the need for electrical isolation. The reliability aspect of such cooling technology has not been well addressed in the open literature. This paper presents the performance of a fully single-phase dielectric fluid immersed server over wide temperature ranges in an environmental chamber. The server was placed in an environmental chamber and applied extreme temperatures ranging from −20 °C to 10 °C at 100% relative humidity and from 20 to 55 °C at constant 50% relative humidity for extended durations. This work is a first attempt of measuring the performance of a server and other components like pump including flow rate drop, starting trouble, and other potential issues under extreme climatic conditions for a completely liquid-submerged system. Pumping power consumption is directly proportional to the operating cost of a data center. The experiment was carried out until the core temperature reached the maximum junction temperature. This experiment helps to determine the threshold capacity and the robustness of the server for its applications in extreme climatic conditions.