From 192f72d61aeb7038166305503b403e2fa902795a Mon Sep 17 00:00:00 2001 From: Dave Friedel Date: Mon, 28 Jul 2025 01:56:50 -0400 Subject: [PATCH] tex formatting tweak --- main.tex | 2 +- ubiquity.pdf | Bin 1967118 -> 1967104 bytes 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/main.tex b/main.tex index ef7c1dd..11e037a 100644 --- a/main.tex +++ b/main.tex @@ -755,7 +755,7 @@ Williams' theoretical result is not merely of academic interest; it describes a By bridging theory and practice, we provide practitioners with concrete guidance on when and how to apply space-time trade-offs. Our open-source tools and complete experimental data (available at \url{https://github.com/sqrtspace}) democratize these optimizations, making theoretical insights accessible for real-world system design. -The ubiquity of the $\sqrt{n}$ pattern---from database buffers to neural network training---validates Williams' mathematical insight. As data continues to grow exponentially while memory grows linearly, understanding and applying these trade-offs becomes increasingly critical for building efficient systems. +The ubiquity of the $\sqrt{n}$ pattern, from database buffers to neural network training, validates Williams' mathematical insight. As data continues to grow exponentially while memory grows linearly, understanding and applying these trade-offs becomes increasingly critical for building efficient systems. \section*{Acknowledgments} This work was carried out independently as part of early-stage R\&D at MarketAlly LLC and MarketAlly Pte. Ltd. The author acknowledges the use of large-language models for drafting, code generation, and formatting assistance. 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