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The article discusses a student research project in which concepts of basic mathematics plays a significant role. It presents sample mathematical exercises depicting induction arguments through the idea of an island having combinatorial problems. It notes that the study offers a one-dimensional case that is convenient for student research.

Abstract: We show that the graph 2-sum of two frameworks is the underlying framework for the 2-sum of the infinitesimal and generic rigidity matroids of the frameworks. However, we show that, unlike the cycle matroid of a graph, these rigidity matroids are not closed under 2-sum decomposition. [Copyright &y& Elsevier]

In our previous paper, we presented the combinatorial theory for minimal isostatic pinned frameworks–Assur graphs–which arise in the analysis of mechanical linkages. In this paper we further explore the geometric properties of Assur graphs, with a focus on singular realizations which have static self-stresses. We provide a new geometric characterization of Assur graphs, based on special singular realizations. These singular positions are then related to dead-end positions in which an associated mechanism with an inserted driver will stop or jam. [Copyright &y& Elsevier]

Abstract: We introduce the idea of Assur graphs, a concept originally developed and exclusively employed in the literature of the kinematics community. This paper translates the terminology, questions, methods and conjectures from the kinematics terminology for one degree of freedom linkages to the terminology of Assur graphs as graphs with special properties in rigidity theory. Exploiting the recent works in combinatorial rigidity theory we provide mathematical characterizations of these graphs derived from ‘minimal’ linkages. With these characterizations, we confirm a series of conjectures posed by Offer Shai, and offer techniques and algorithms to be exploited further in future work. [Copyright &y& Elsevier]

The article discusses the application of Burnside's Lemma in solving a one-dimensional Escher problem of contemporary combinatorics in mathematical analysis. The problem is simplified through different patterns by repeatedly translating a 2x2 square, it has four unit squares filled with asymmetric motif in randomly rotated aspects. The result of the simplification represents a mathematical approach in permutation.

Focuses on the solution on the n-jugs and water problem using Euclidean algorithm. Variable used on n-jugs; Use of variable d as the greatest common divisor of variable L; Importance of getting the mathematical product of d in the solution.