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Generalizing is a critical aspect of mathematics learning, with researchers and policy documents highlighting generalizing as a core mathematical practice. It can also be challenging to foster in class settings, and teachers need access to better resources to teach generalizing, including an understanding of effective forms of instruction. This article proposes Classroom Supports for Generalizing (CSGs), investigating how multiple elements—such as tasks, teacher moves, student interactions, and representations—interact to meaningfully foster student generalizing. Drawing on class video data from a middle school teacher and two high school teachers, we present the CSG Framework, which identifies three categories of supports: Interactions for Generalizing, Structures for Generalizing, and Routines for Generalizing. 

Printed electronics are gaining significant interest due to their design flexibility, low fabrication cost, and rapid design-to-manufacturing turnaround. Conventional substrates for printed electronics are often based on nonbiodegradable polymers such as polyimide that pose high environmental challenges by creating massive e-waste and pollution. As the demand for printed electronics and sensors increases, the ability to print such devices on biodegradable substrates can provide a solution to such environmental problems. However, current printing technologies are based on liquids and inks that are incompatible with biodegradable substrates, such as paper. Here, we present a dry-printing process, namely, a dry additive nanomanufacturing (Dry-ANM) technique, for printing conductive silver lines and patterns on biodegradable papers for flexible hybrid papertronics. Pure and dry nanoparticles are generated by pulsed laser ablation of a silver target that is then transported through a nozzle and directed onto paper substrates, where they are deposited and laser-sintered in real time to form the desired pattern without damaging the paper. The effects of different printing parameters on the paper-burning threshold are investigated, and the electrical properties of the lines are characterized by using different line thicknesses and sintering laser power densities. In addition, the mechanical and electrical properties of the printed lines and patterns are evaluated by bending and twisting tests. Furthermore, the feasibility of printing silver on different paper types is demonstrated. This research can potentially lead to biodegradable and environmentally friendly printed electronics and sensors. 

The MYC oncogenic transcription factor is acetylated by the p300 and GCN5 histone acetyltransferases. The significance of MYC acetylation and the functions of specific acetylated lysine (AcK) residues have remained unclear. Here, we show that the major p300-acetylated K148(149) and K157(158) sites in human (or mouse) MYC and the main GCN5-acetylated K323 residue are reversibly acetylated in various malignant and nonmalignant cells. Oncogenic overexpression of MYC enhances its acetylation and alters the regulation of site-specific acetylation by proteasome and deacetylase inhibitors. Acetylation of MYC at different K residues differentially affects its stability in a cell type-dependent manner. Lysine-to-arginine substitutions indicate that although none of the AcK residues is required for MYC stimulation of adherent cell proliferation, individual AcK sites have gene-specific functions controlling select MYC-regulated processes in cell adhesion, contact inhibition, apoptosis, and/or metabolism and are required for the malignant cell transformation activity of MYC. Each AcK site is required for anchorage-independent growth of MYC-overexpressing cells in vitro, and both the AcK148(149) and AcK157(158) residues are also important for the tumorigenic activity of MYC transformed cells in vivo. The MYC AcK site-specific signaling pathways identified may offer new avenues for selective therapeutic targeting of MYC oncogenic activities. 

Volcanic highlands in the tropics increased global weatherability ~0.9–1.6 Ma before onset of Cryogenian Snowball Earth. 

Increased availability of high-quality customer information has fueled interest in personalized pricing strategies, that is, strategies that predict an individual customer’s valuation for a product and then offer a price tailored to that customer. Although the appeal of personalized pricing is clear, it may also incur large costs in the forms of market research, investment in information technology and analytics expertise, and branding risks. In light of these trade-offs, our work studies the value of personalized pricing strategies over a simple single-price strategy. We first provide closed-form lower and upper bounds on the ratio between the profits of an idealized personalized pricing strategy (first-degree price discrimination) and a single-price strategy. Our bounds depend on simple statistics of the valuation distribution and shed light on the types of markets for which personalized pricing has little or significant potential value. Second, we consider a feature-based pricing model where customer valuations can be estimated from observed features. We show how to transform our aforementioned bounds into lower and upper bounds on the value of feature-based pricing over single pricing depending on the degree to which the features are informative for the valuation. Finally, we demonstrate how to obtain sharper bounds by incorporating additional information about the valuation distribution (moments or shape constraints) by solving tractable linear optimization problems. This paper was accepted by David Simchi-Levi, revenue management and market analytics. 

We study the power of selling opaque products, that is, products where a feature (such as color) is hidden from the customer until after purchase. Opaque products, which are sold with a price discount, have emerged as a powerful vehicle to increase revenue for many online retailers and service providers that offer horizontally differentiated items. In the opaque selling models we consider, all of the items are sold at a single common price alongside opaque products that may correspond to various subsets of the items. We consider two types of customers, risk-neutral ones, who assume they will receive a truly random item of the opaque product, and pessimistic ones, who assume they will receive their least favorite item of the opaque product. We benchmark opaque selling against two common selling strategies: discriminatory pricing, where one explicitly charges different prices for each item, and single pricing, where a single price is charged for all the items. We give a sharp characterization of when opaque selling outperforms discriminatory pricing; namely, this result holds for situations where all customers are pessimistic or the item valuations are supported on two points. In the latter case, we also show that opaque selling with just one opaque product guarantees at least 71.9% of the revenue from discriminatory pricing. We then provide upper bounds on the potential revenue increase from opaque selling strategies over single pricing and describe cases where the increase can be significantly more than that of discriminatory pricing. Finally, we provide pricing algorithms and conduct an extensive numerical study to assess the power of opaque selling for a variety valuation distributions and model extensions. This paper was accepted by Gabriel Weintraub, revenue management and market analytics. 

In the online video game industry, a significant portion of the revenue is generated from microtransactions, where a small amount of real-world currency is exchanged for virtual items to be used in the game. One popular way to conduct microtransactions is via a loot box, which is a random allocation of virtual items whose contents are not revealed until after purchase. In this work, we consider how to optimally price and design loot boxes from the perspective of a revenue-maximizing video game company and analyze customer surplus under such selling strategies. Our paper provides the first formal treatment of loot boxes, with the aim to provide customers, companies, and regulatory bodies with insights into this popular selling strategy. We consider two types of loot boxes: a traditional one where customers can receive (unwanted) duplicates and a unique one where customers are guaranteed to never receive duplicates. We show that as the number of virtual items grows large, the unique box strategy is asymptotically optimal among all possible strategies, whereas the traditional box strategy only garners 36.7% of the optimal revenue. On the other hand, the unique box strategy leaves almost zero customer surplus, whereas the traditional box strategy leaves positive surplus. Further, when designing traditional and unique loot boxes, we show it is asymptotically optimal to allocate the items uniformly, even when the item valuation distributions are heterogeneous. We also show that, when the seller purposely misrepresents the allocation probabilities, their revenue may increase significantly, and thus, strict regulation is needed. Finally, we show that, even if the seller allows customers to salvage unwanted items, then the customer surplus can only increase by at most 1.4%. This paper was accepted by Victor Martinez-de-Albeniz, operations management.