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Abstract. Considering any set of n-positive masses, n ≥ 3, moving in R 2 under Newtonian gravitation, we prove that action-minimizing solutions in the class of paths with rotational and reflection symmetries are collision-free. For an open set of masses, the periodic and quasi-periodic solutions we obtained contain and extend the classical EulerMoulton relative equilibria. We also show several numerical results on these actionminimizing solutions. Using a natural topological classification for collision-free paths via their braid types in a rotating frame, these action-minimizing solutions change from trivial to non-trivial braids as we vary masses and other parameters.

Rationale: Computerized tomography (CT)-guided blue dye localization has been widely discussed for preoperative localization of pulmonary nodules. However, few studies have investigated this technique for intra-abdominal lesions. Although preoperative localization is not commonly required in laparotomy, it may assume importance with advancements in the field of laparoscopic surgery. Patient concerns: Herein, we report the cases of 2 patients diagnosed with colon cancer who underwent hemicolectomy with extended lymphadenectomy and subsequent chemotherapy. Diagnoses: Follow-up CT scans showed newly developed metastatic lymphadenopathy and peritoneal tumor implants. Interventions: Considering the difficulty in identification of and access to the target lesions during laparoscopic surgery, preoperative CT-guided blue dye localization was performed in both cases. Outcomes: All the target lesions were identified by the dye marker and removed successfully. The pathologic results revealed adenocarcinoma. Lessons: We established the following strategy for preoperative CT-guided dye localization of intra-abdominal lesions: Intra-abdominal lesions that are hard to identify due to their size or morphology, and difficult to approach due to their location or surrounding structures, maybe the candidates for this procedure, especially in cases of laparoscopic surgery. Operators should adjust their localization planning based on the surgery method, cutting path, and location of port sites. The target dye marker should be clearly visible in the presumed intra-operative field of view. A second dye marker should be made to ensure surgical success when the target dye marker is obscured by the surrounding structures in the presumed intra-operative field of view.

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