LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Modelling and Simulation for Power Distribution Grids of 3D Tiled Computing Arrays

This thesis presents modelling and simulation developments for power distribution grids of 3D tiled computing arrays (TCAs), a novel type of paradigm for HPC systems, and tests the feasibility of such systems for HPC systems domains. The exploration of a complex power-grid such as those found in the TCA concept requires detailed simulations of systems with hundreds and possibly thousands of modular nodes, each contributing to the collective behaviour of the system. In particular power, voltage, and current behaviours are critically important observations. To facilitate this investigation, and test the hypothesis, which seeks to understand if scalability is feasible for such systems, a bespoke simulation platform has been developed, and (importantly) validated against hardware prototypes of small systems. A number of systems are simulated, including systems consisting of arrays of ’balls’. Balls are collections of modular tiles that form a ball-like modular unit, and can then themselves be tiled into large scale systems. Evaluations typically involved simulation of cubic arrays of sizes ranging from 2x2x2 balls up to 10x10x10. Larger systems require extended simulation times. Therefore models are developed to extrapolate system behaviours for higher-orders of systems and to gauge the ultimate scalability of such TCA systems. It is found that systems of 40x40x40 are quite feasible with appropriate configurations. Data connectivity is explored to a lesser degree, but comparisons were made between TCA systems and well known comparable HPC systems, and it is concluded that TCA systems can be built with comparable data-flow and scalability, and that the electrical and engineering challenges associated with the novelty of 3D tiled systems can be met with practical solutions

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2