Mathematical Sciences- Journal Articles
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Item Dynamics of a time-delayed relay system(American Physical Society, 2024-01-23) Illing, Lucas; Ryan, Pierce; Amann, Andreas; Irish Research Council; McAfeeWe study the dynamics of a piecewise-linear second-order delay differential equation that is representative of feedback systems with relays (switches) that actuate after a fixed delay. The system under study exhibits strong multirhythmicity, the coexistence of many stable periodic solutions for the same values of the parameters. We present a detailed study of these periodic solutions and their bifurcations. Starting from an integrodifferential model, we show how to reduce the system to a set of finite-dimensional maps. We then demonstrate that the parameter regions of existence of periodic solutions can be understood in terms of discontinuity-induced bifurcations and their stability is determined by smooth bifurcations. Using this technique, we are able to show that slowly oscillating solutions are always stable if they exist. We also demonstrate the coexistence of stable periodic solutions with quasiperiodic solutions.Item Does correction of carpal malalignment influence the union rate of scaphoid nonunion surgery?(SAGE Publishing, 2023-11-21) McLaughlin, Kealan; Jabbar, Faizan A. A.; Kelly, Luke J.; Jovanovic, Iva; Gray, Matthew P.; Charalambous, Charalambos P.; Harrison, John W. K.The aim of this retrospective study was to assess the relation between carpal malalignment correction and radiological union rates in surgery for scaphoid nonunions. A total of 59 scaphoid waist fracture nonunions treated with open reduction and palmar tricortical autograft were divided according to their pre- and postoperative scapholunate (SL) and radiolunate (RL) angles. We found that carpal malalignment failed to correct in 32 of 59 (54.2%) patients despite meticulous surgical technique and placement of an appropriately sized wedge-shaped graft. In total, 43 (72.9%) fractures united at a mean of 4.47 months (range 3–11). Of the 27 fractures with postoperative SL and RL angles within the normal range, 21 united, whereas 22 of the 32 remaining fractures that failed to achieve postoperative angles within the normal range went on to union. The postoperative SL and RL angles were not related to union. Our findings suggest that in scaphoid fracture nonunion surgery, carpal malalignment may not be corrected in a substantial proportion of patients, but such correction may not be essential for bony union. Our findings also show that there is no marked collapse of the scaphoid graft in the early postoperative period.Item Constant mean curvature surfaces of any positive genus(Cambridge University Press, 2005-07-20) Kilian, Martin; Kobayashi, S.-P.; Rossman , W.; Schmitt, N.; Ministry of Education, Culture, Sports, Science and Technology; Deutsche Forschungsgemeinschaft; Engineering and Physical Sciences Research CouncilThe paper shows the existence of several new families of noncompact constant mean curvature surfaces: (i) singly punctured surfaces of arbitrary genus g 1, (ii) doubly punctured tori, and (iii) doubly periodic surfaces with Delaunay ends.Item Dressing preserving the fundamental group(Elsevier B.V., 2005-07-14) Dorfmeister, J.; Kilian, Martin; Engineering and Physical Sciences Research CouncilIn this note we consider the relationship between the dressing action and the holonomy representation in the context of constant mean curvature surfaces. We characterize dressing elements that preserve the topology of a surface and discuss dressing by simple factors as a means of adding bubbles to a class of non-finite type cylinders.Item Constant mean curvature trinoids with one irregular end(Tohoku University, Mathematical Institute, 2020-06-26) Kilian, Martin; Mota, Eduardo; Schmitt, NicholasWe construct a new five parameter family of constant mean curvature trinoids with two asymptotically Delaunay ends and one irregular end.