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1. Signed area enumeration for lattice paths

   Polychronakos, Alexios P; Ouvry, Stephane (April 2023, Séminaire lotharingien de combinatoire)

   We give a summary of recent progress on the signed area enumeration of 9 closed walks on planar lattices. Several connections are made with quantum mechanics 10 and statistical mechanics. Explicit combinatorial formulae are proposed which rely on 11 sums labelled by the multicompositions of the length of the walks. 

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2. Signed area enumeration for lattice paths

   Stephane, Ouvry; Polychronakos, Alexios P. (April 2023, Séminaire lotharingien de combinatoire)

   We give a summary of recent progress on the signed area enumeration of 9 closed walks on planar lattices. Several connections are made with quantum mechanics 10 and statistical mechanics. Explicit combinatorial formulae are proposed which rely on 11 sums labelled by the multicompositions of the length of the walks. 

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3. Rising in the ranks!: learning math or playing games?

   https://doi.org/10.3389/feduc.2023.1302693  

   Sager, Marc T; Sherard, Maximilian K; Milton, Saki; Walkington, Candace; Petrosino, Anthony J (December 2023, Frontiers in Education)

   This qualitative case study examines the learning that occurred when a small group of middle grade youths embarked upon a personal excursion during a game-based math walk. Math walks are an informal learning activity where learners and facilitators explore mathematical concepts embedded in everyday spaces. The MathExplorer app is a location-based mobile game designed to enhance and gamify math walks. In our broader research, we investigated a group of 18 middle grade learners who used MathExplorer to engage in math walks at a local nature preserve. While most youths in this study used the game as planned by the researchers, one group deviated from the plan and devised new ways of playing the game and participating in the math walks. We see this deviation, or personal excursion, as a source of insight for research on game-based math walks. To understand the learning that took place during this personal excursion, we draw upon sociocultural and self-directed theories of learning. Using methods of interaction analysis and embodied action conversation framework, we analyzed the small groups’ discussion, movement, and game-use to understand: (1) the point at which the students departed from the planned use of MathExplorer; and (2) the learning that took place after this departure. The findings include how the youth explicitly incorporate mathematics into game play through an activity-as-planned, and how the youth embark on a personal excursion relating to game mechanics and gamification, with an implicit focus on mathematics. We discuss the importance of personal excursions for designing informal mathematics learning experiences. 

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4. Algebraic area enumeration for open lattice walks

   https://doi.org/10.1088/1751-8121/aca573  

   Ouvry, Stéphane; Polychronakos, Alexios P (December 2022, Journal of Physics A: Mathematical and Theoretical)

   Abstract We calculate the number of open walks of fixed length and algebraic area on a square planar lattice by an extension of the operator method used for the enumeration of closed walks. The open walk area is defined by closing the walks with a straight line across their endpoints and can assume half-integer values in lattice cell units. We also derive the length and area counting of walks with endpoints on specific straight lines and outline an approach for dealing with walks with fully fixed endpoints. 

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5. Transport of complex and active fluids in porous media

   https://doi.org/10.1122/8.0000389  

   Kumar, Manish; Guasto, Jeffrey S.; Ardekani, Arezoo M. (March 2022, Journal of Rheology)

   Complex and active fluids find broad applications in flows through porous materials. Nontrivial rheology can couple to porous microstructure leading to surprising flow patterns and associated transport properties in geophysical, biological, and industrial systems. Viscoelastic instabilities are highly sensitive to pore geometry and can give rise to chaotic velocity fluctuations. A number of recent studies have begun to untangle how the pore-scale geometry influences the sample-scale flow topology and the resulting dispersive transport properties of these complex systems. Beyond classical rheological properties, active colloids and swimming cells exhibit a range of unique properties, including reduced effective viscosity, collective motion, and random walks, that present novel challenges to understanding their mechanics and transport in porous media flows. This review article aims to provide a brief overview of essential, fundamental concepts followed by an in-depth summary of recent developments in this rapidly evolving field. The chosen topics are motivated by applications, and new opportunities for discovery are highlighted. 

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