Let's hope your couch is shaped like an old-fashioned telephone receiver. A graphical representation of what a real Gerver's sofa might look like. Breakthroughs, discoveries, and DIY tips sent every weekday. If you've ever struggled to squeeze a couch around a tight corner while moving into a new apartment, you'll probably find that the pure mathematics problem known as the "sofa problem" is incredibly relatable. The question seeks to find a maximum value for the area of a sofa that can slide around a 90-degree corner in a corridor of a given width.

The Moving Sofa Problem, formally proposed by Leo Moser in 1966, seeks to determine the largest area of a two-dimensional shape that can navigate through an $L$-shaped corridor with unit width. The current best lower bound is about 2.2195, achieved by Joseph Gerver in 1992, though its global optimality remains unproven. In this paper, we investigate this problem by leveraging the universal approximation strength and computational efficiency of neural networks. We report two approaches, both supporting Gerver's conjecture that his shape is the unique global maximum. Our first approach is continuous function learning. We drop Gerver's assumptions that i) the rotation of the corridor is monotonic and symmetric and, ii) the trajectory of its corner as a function of rotation is continuously differentiable. We parameterize rotation and trajectory by independent piecewise linear neural networks (with input being some pseudo time), allowing for rich movements such as backward rotation and pure translation. We then compute the sofa area as a differentiable function of rotation and trajectory using our "waterfall" algorithm. Our final loss function includes differential terms and initial conditions, leveraging the principles of physics-informed machine learning. Under such settings, extensive training starting from diverse function initialization and hyperparameters is conducted, unexceptionally showing rapid convergence to Gerver's solution. Our second approach is via discrete optimization of the Kallus-Romik upper bound, which converges to the maximum sofa area from above as the number of rotation angles increases. We uplift this number to 10000 to reveal its asymptotic behavior. It turns out that the upper bound yielded by our models does converge to Gerver's area (within an error of 0.01% when the number of angles reaches 2100). We also improve their five-angle upper bound from 2.37 to 2.3337.

We outline an unsupervised method for temporal rank ordering of sets of historical documents, namely American State of the Union Addresses and DEEDS, a corpus of medieval English property transfer documents. Our method relies upon effectively capturing the gradual change in word usage via a bandwidth estimate for the non-parametric Generalized Linear Models (Fan, Heckman, and Wand, 1995). The number of possible rank orders needed to search through possible cost functions related to the bandwidth can be quite large, even for a small set of documents. We tackle this problem of combinatorial optimization using the Simulated Annealing algorithm, which allows us to obtain the optimal document temporal orders. Our rank ordering method significantly improved the temporal sequencing of both corpora compared to a randomly sequenced baseline. This unsupervised approach should enable the temporal ordering of undated document sets.

The industrial advancements of the past -- from machines enabling mass production to the introduction of computers and automation -- have all led to the tipping point we are navigating today. Organizations of all sizes and across sectors are increasingly adopting advanced digital technologies. This will transform the way we work as drastically as past achievements did a century ago. Leaders need to ensure their organizations can seamlessly navigate these shifts and have the necessary resources to drive the promise of digitization forward. While businesses had some time to prepare for digital transformation, the pandemic propelled those efforts.

In a move that could transform manuscript studies, University of Toronto researchers have partnered with a team in the United Kingdom to develop a program that can read and transcribe the handwritten Latin found in 13th-century legal manuscripts. While scholars have been making digital images of these manuscripts for years, transcribing and comparing these texts is painstaking and tedious work that can take years or even decades to complete. That's because medieval handwriting can often look crabbed and unintelligible, with non-standardized spellings, hyphenations, abbreviations, calligraphic flourishes and any number of distinct "hands." But machine-reading software called Transkribus promises to change the field. Using artificial intelligence (AI), the software can theoretically be trained to read any type of handwriting, in any language – and Michael Gervers, a professor of medieval social and economic history at U of T Scarborough, says it could eventually be applied across medieval studies.