We show how frictions and continuous transfers jointly affect equilibria in a model of matching in trading networks. Our model incorporates distortionary frictions such as transaction taxes and commissions. When contracts are fully substitutable for firms, competitive equilibria exist and coincide with outcomes that satisfy a cooperative solution concept called trail stability. However, competitive equilibria are generally neither stable nor Pareto‐efficient.

Novel binary gene expression tools like the LexA-LexAop system could powerfully enhance studies of metabolism, development, and neurobiology in Drosophila. However, specific LexA drivers for neuroendocrine cells and many other developmentally relevant systems remain limited. In a unique high school biology course, we generated a LexA-based enhancer trap collection by transposon mobilization. The initial collection provides a source of novel LexA-based elements that permit targeted gene expression in the corpora cardiaca, cells central for metabolic homeostasis, and other neuroendocrine cell types. The collection further contains specific LexA drivers for stem cells and other enteric cells in the gut, and other developmentally relevant tissue types. We provide detailed analysis of nearly 100 new LexA lines, including molecular mapping of insertions, description of enhancer-driven reporter expression in larval tissues, and adult neuroendocrine cells, comparison with established enhancer trap collections and tissue specific RNAseq. Generation of this open-resource LexA collection facilitates neuroendocrine and developmental biology investigations, and shows how empowering secondary school science can achieve research and educational goals.

We study the central hyperplane arrangement whose hyperplanes are the vanishing loci of the weights of the first and the second fundamental representations of gl n restricted to the dual fundamental Weyl chamber. We obtain generating functions that count flats and faces of a given dimension. This counting is interpreted in physics as the enumeration of the phases of the Coulomb and mixed Coulomb-Higgs branches of a five dimensional gauge theory with 8 supercharges in presence of hypermultiplets transforming in the fundamental and antisymmetric representation of a U (n) gauge group as described by the Intriligator-Morrison-Seiberg superpotential.2010 Mathematics Subject Classification. 05E10, 52C35, 05A15, 17B10, 17B81.

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We investigate pattern avoidance in alternating permutations and generalizations thereof. First, we study pattern avoidance in an alternating analogue of Young diagrams. In particular, we extend Babson-West's notion of shape-Wilf equivalence to apply to alternating permutations and so generalize results of Backelin-West-Xin and Ouchterlony to alternating permutations. Second, we study pattern avoidance in the more general context of permutations with restricted ascents and descents. We consider a question of Lewis regarding permutations that are the reading words of thickened staircase Young tableaux, that is, permutations that have k − 1 ascents followed by a descent, followed by k − 1 ascents, et cetera. We determine the relative sizes of the sets of pattern-avoiding (k − 1)-ascent permutations in terms of the forbidden pattern. Furthermore, inequalities in the sizes of sets of pattern-avoiding permutations in this context arise from further extensions of shape-equivalence type enumerations.Definition 3.1. Let Y be a Young diagram with k rows. If A and D are disjoint subsets of [k − 1] such that if i ∈ A ∪ D, then the ith and (i + 1)st rows of Y have the same length, then we call the triple Y = (Y, A, D) an AD-Young diagram. We call Y the Young diagram of Y, A the required ascent set of Y, and D the required descent set of Y. See Figure 2.As in [1,2,14], a transversal of Young diagram Y is a set of squares T = {(i, t i )} such that every row and every column of Y contains exactly one member of T .We call Asc(T ) the ascent set of T and Des(T ) the descent set of T . If A ⊆ A ′ and D ⊆ D ′ , then we say that T a valid transversal of Y. Example 3.3. If T is a transversal of a Young diagram Y , then T is a valid transversal of the AD-Young diagram (Y, ∅, ∅).Except for a brief digression in Section 7, we restrict ourselves to the AD-Young analogues of alternating and reverse alternating permutations. Definition 3.4. Given positive integers x, y and an AD-Young diagram (Y, A, D) such that Y has k rows, we say that (Y, A, D) is x, y-alternating if A, D satisfy the property that if x − 1 ≤ i ≤ k − y, then i ∈ A if and only if i + 1 ∈ D.Example 3.5. Let Y = (4 4 ). Then, (Y, {1}, {2}) is 1-alternating, while (Y, {1, 3}, {2}) is 2-alternating but not 1-alternating. Furthermore, (Y, {2, 4}, {1, 3}) is 1-semialternating but not y-alternating for y ≤ 4.The notion of pattern avoidance is exactly as in [1,2,14]; if a transversal T = {(i, t i )} of a Young diagram Y contains a r×r permutation matrix M if there are rows a 1 < a 2 < · · · < a r and columns b 1 < b 2 < · · · < b r of Y such that (a r , b r ) ∈ Y and the restriction of T to the rows a i and the columns b i has 1's exactly where M has 1's. If T does not contain M, then we say that T avoids M. See Figure 3. Given an AD-Young diagram Y and a permutation matrix M, let S Y (M) denote the set of valid transversals of Y that avoid M. Definition 3.6. If M and N are permutation matrices such that |S Y (M)| = |S Y (N)| for all x-alternating AD-Young diagrams Y, we say that...

Biological materials, such as proteins, often have a hierarchical structure ranging from basic building blocks at the nanoscale (e.g., amino acids) to assembled structures at the macroscale (e.g., fibers). Current software for materials engineering allows the user to specify polypeptide chains and simple secondary structures prior to molecular dynamics simulation, but is not flexible in terms of the geometric arrangement of unequilibrated structures. Given some knowledge of a larger-scale structure, instructing the software to create it can be very difficult and time-intensive. To this end, the present paper reports a mathematical language, using category theory, to describe the architecture of a material, i.e., its set of building blocks and instructions for combining them. While this framework applies to any hierarchical material, here we concentrate on proteins. We implement this mathematical language as an open-source Python library called Matriarch. It is a domain-specific language that gives the user the ability to create almost arbitrary structures with arbitrary amino acid sequences and, from them, generate Protein Data Bank (PDB) files. In this way, Matriarch is more powerful than commercial software now available. Matriarch can be used in tandem with molecular dynamics simulations and helps engineers design and modify biologically inspired materials based on their desired functionality. As a case study, we use our software to alter both building blocks and building instructions for tropocollagen, and determine their effect on its structure and mechanical properties.

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We show how frictions and continuous transfers jointly affect equilibria in a model of matching in trading networks. Our model incorporates distortionary frictions such as transaction taxes and commissions. When contracts are fully substitutable for firms, competitive equilibria exist and coincide with outcomes that satisfy a cooperative solution concept called trail stability. However, competitive equilibria are generally neither stable nor Pareto-efficient.