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1. Weighted Mobility

   https://doi.org/10.1002/adma.202001537  

   Snyder, G_Jeffrey; Snyder, Alemayouh_H; Wood, Maxwell; Gurunathan, Ramya; Snyder, Berhanu_H; Niu, Changning (May 2020, Advanced Materials)

   Abstract Engineering semiconductor devices requires an understanding of charge carrier mobility. Typically, mobilities are estimated using Hall effect and electrical resistivity meausrements, which are are routinely performed at room temperature and below, in materials with mobilities greater than 1 cm²V^‐1s^‐1. With the availability of combined Seebeck coefficient and electrical resistivity measurement systems, it is now easy to measure the weighted mobility (electron mobility weighted by the density of electronic states). A simple method to calculate the weighted mobility from Seebeck coefficient and electrical resistivity measurements is introduced, which gives good results at room temperature and above, and for mobilities as low as 10⁻³cm²V^‐1s^‐1,Here, μ_wis the weighted mobility, ρ is the electrical resistivity measured in mΩ cm,Tis the absolute temperature in K,Sis the Seebeck coefficient, andk_B/e = 86.3 µV K^–1. Weighted mobility analysis can elucidate the electronic structure and scattering mechanisms in materials and is particularly helpful in understanding and optimizing thermoelectric systems. 

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2. Weighted Brunn-Minkowski theory I: On weighted surface area measures

   Fradelizi, M; Langharst, D; Madiman, M; Zvavitch, A (February 2024, Journal of mathematical analysis and applications)
3. Weighted Alpert Wavelets

   https://doi.org/10.1007/s00041-020-09784-0  

   Rahm, Rob; Sawyer, Eric T.; Wick, Brett D. (February 2021, Journal of Fourier Analysis and Applications)

   null (Ed.)
4. Weighted homological regularities

   https://doi.org/10.1090/tran/8997  

   Kirkman, E.; Won, R.; Zhang, J. (August 2023, Transactions of the American Mathematical Society)

   Dan Abramovich (Ed.)

   Let $A$be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the homological and internal degrees of cochain complexes of graded $A$-modules, providing weighted versions of Castelnuovo–Mumford regularity, Tor-regularity, Artin–Schelter regularity, and concavity. In some cases an invariant (such as Tor-regularity) that is infinite can be replaced with a weighted invariant that is finite, and several homological invariants of complexes can be expressed as weighted homological regularities. We prove a few weighted homological identities some of which unify different classical homological identities and produce interesting new ones. 

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5. Weighted Additive Spanners

   https://doi.org/10.1007/978-3-030-60440-0_32  

   Ahmed, R; Bodwin, G; Darabi, F; Kobourov, S; Spence, R (October 2020, 46th International Workshop on Graph-Theoretic Concepts in Computer Science (WG))

   A spanner of a graph G is a subgraph H that approximately preserves shortest path distances in G. Spanners are commonly applied to compress computation on metric spaces corresponding to weighted input graphs. Classic spanner constructions can seamlessly handle edge weights, so long as error is measured multiplicatively. In this work, we investigate whether one can similarly extend constructions of spanners with purely additive error to weighted graphs. These extensions are not immediate, due to a key lemma about the size of shortest path neighborhoods that fails for weighted graphs. Despite this, we recover a suitable amortized version, which lets us prove direct extensions of classic +2 and +4 unweighted spanners (both all-pairs and pairwise) to +2W and +4W weighted spanners, where W is the maximum edge weight. Specifically, we show that a weighted graph G contains all-pairs (pairwise) +2W and +4W weighted spanners of size O(n3/2) and O(n7/5) (O(np1/3) and O(np2/7)) respectively. For a technical reason, the +6 unweighted spanner becomes a +8W weighted spanner; closing this error gap is an interesting remaining open problem. That is, we show that G contains all-pairs (pairwise) +8W weighted spanners of size O(n4/3) (O(np1/4)). 

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