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1. Party System Congruence and Bicameralism

   https://doi.org/10.1177/00104140241302767  

   Kim, Wooseok; Hicken, Allen; Kollman, Ken (November 2024, Comparative Political Studies)

   We compare the nature of party systems across bicameral legislatures using newly available data on upper chamber elections. We examine the similarity in the composition of political parties between the lower and upper chambers (partisan congruence) and introduce a novel measure that captures differences in the nationalization of parties between the two chambers (nationalization congruence). We explore variations in these measures across countries and over time and demonstrate that the power of the upper chamber (symmetry) is linked to both forms of congruence. Moreover, we apply these measures to understand how the interaction between congruence and symmetry—two key dimensions of bicameralism—influences policymaking, focusing on government spending patterns. Our findings reveal that partisan and nationalization congruence can have contrasting implications for government spending in symmetric bicameral systems but have negligible implications in asymmetric bicameral systems. 

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2. Modularity and Combination of Associative Commutative Congruence Closure Algorithms enriched with Semantic Properties

   https://doi.org/10.46298/lmcs-19(1:19)2023  

   Kapur, Deepak (March 2023, Logical Methods in Computer Science)

   Algorithms for computing congruence closure of ground equations overuninterpreted symbols and interpreted symbols satisfying associativity andcommutativity (AC) properties are proposed. The algorithms are based on aframework for computing a congruence closure by abstracting nonflat terms byconstants as proposed first in Kapur's congruence closure algorithm (RTA97).The framework is general, flexible, and has been extended also to developcongruence closure algorithms for the cases when associative-commutativefunction symbols can have additional properties including idempotency,nilpotency, identities, cancellativity and group properties as well as theirvarious combinations. Algorithms are modular; their correctness and terminationproofs are simple, exploiting modularity. Unlike earlier algorithms, theproposed algorithms neither rely on complex AC compatible well-foundedorderings on nonvariable terms nor need to use the associative-commutativeunification and extension rules in completion for generating canonical rewritesystems for congruence closures. They are particularly suited for integratinginto the Satisfiability modulo Theories (SMT) solvers. A new way to viewGroebner basis algorithm for polynomial ideals with integer coefficients as acombination of the congruence closures over the AC symbol * with the identity 1and the congruence closure over an Abelian group with + is outlined. 

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3. Prospective High School Teachers’ Understanding and Application of the Connection Between Congruence and Transformation in Congruence Proofs

   St. Goar, J.; Lai, Y.; Funk, R. (January 2019, Proceedings of the 22nd Annual Conference on Research in Undergraduate Mathematics Education)

   Undergraduate mathematics instructors are called by recent standards to promote prospective teachers’ learning of a transformation approach in geometry and its proofs. The novelty of this situation means it is unclear what is involved in prospective teachers’ learning of geometry from a transformation perspective, particularly if they learned geometry from an approach based on the Elements; hence undergraduate instructors may need support in this area. To begin to approach this problem, we analyze the prospective teachers’ use of the conceptual link between congruence and transformation in the context of congruence. We identify several key actions involved in using the definition of congruence in congruence proofs, and we look at ways in which several of these actions are independent of each other, hence pointing to concepts and actions that may need to be specifically addressed in instruction. 

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4. Negative frequency-dependent selection contributes to modular structure of effector repertoires in Pseudomonas syringae

   https://doi.org/10.1101/2025.01.16.632277  

   Li, Shuanger; Laderman, Eric; Märkle, Hanna; Yang, Yunze; Bergelson, Joy; Pascual, Mercedes (January 2025, bioRxiv)

   Abstract The evolutionary fate of multi-strain pathogens is shaped by host-pathogen ecological interactions. In bacterial pathogens of plants, enhanced strain characterization and advances in our understanding of molecular mechanisms underlying defense pathways open the door for revisiting the role of negative frequency-dependent selection (NFDS) in strain structure, including its interplay with genetic exchange. NFDS arising from specific defense is one potential mechanism for generating, maintaining, and structuring pathogen diversity. In plants, specific protection against microbial pathogens involves Resistance proteins (R-proteins) that recognize virulence factors (effectors) secreted by pathogens, typically to subvert the initial line of host defense. Here we formulate a stochastic computational co-evolution model that explicitly incorporates variable length R-gene and effector repertoires, and migration from their regional pools. We use this model to understand potential mechanisms shaping effector repertoire structure and associated strain coexistence in the generalist plant pathogenP. syringae. The demonstration of a modular structure in our numerical simulations motivates the analysis of genome sequences from 76 strains collected in the Midwestern US and 1104 strains from global sources. We find that effector repertories both locally and globally exhibit a modular structure, with higher similarity within than between clusters. The observed modules are consistent with the core genome phylogeny and are unexplained by plant host species, location of isolation, and genetic linkage between effectors. An extension of the model is needed to take into account the evidence for genetic exchange and the phylogenetic congruence of effector modules. We initialize the system with a phylogenetically congruent modular structure and include recombination rates decreasing as a function of phylogenetic distance. We show that NFDS can counter-balance the effects of mixing due to recombination and in so doing, contributes to the maintenance of strain structure. These findings indicate that the observed similarity clusters may constitute, in part, emergent niches arising from eco-evolutionary dynamics that contribute to strain coexistence. 

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5. A Modular Associative Commutative (AC) Congruence Closure Algorithm

   https://doi.org/10.4230/lipics.fscd.2021.15  

   Kapur, Deepak (January 2021, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Kobayashi, Naoki (Ed.)

   Algorithms for computing congruence closure of ground equations over uninterpreted symbols and interpreted symbols satisfying associativity and commutativity (AC) properties are proposed. The algorithms are based on a framework for computing the congruence closure by abstracting nonflat terms by constants as proposed first in Kapur’s congruence closure algorithm (RTA97). The framework is general, flexible, and has been extended also to develop congruence closure algorithms for the cases when associative-commutative function symbols can have additional properties including idempotency, nilpotency and/or have identities, as well as their various combinations. The algorithms are modular; their correctness and termination proofs are simple, exploiting modularity. Unlike earlier algorithms, the proposed algorithms neither rely on complex AC compatible well-founded orderings on nonvariable terms nor need to use the associative-commutative unification and extension rules in completion for generating canonical rewrite systems for congruence closures. They are particularly suited for integrating into Satisfiability modulo Theories (SMT) solvers. 

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