المؤلفون: SEUNG-YEAL HA^1,2 syha@snu.ac.kr, HANSOL PARK³

المصدر: SIAM Journal on Applied Mathematics. 2021, Vol. 81 Issue 5, p2130-2152. 23p.

مصطلحات موضوعية: *DYNAMICAL systems, *POLYTOPES, *LINEAR dynamical systems, *GEOMETRIC approach, *SYSTEM dynamics

مستخلص: We present a dynamical systems approach for geometric matchings in an ensemble of polytopes along rigid-body motions. Each polytope can be characterized by a vertex set and edge or faces determined by vertices, and polygons and simplexes correspond to a polytope. For a geometric matching, we propose a system of dynamical system for the evolution of centroids and rotations of polytopes to match the vertices under rigid-body motions which can be decomposed as a composition of translation and rotations. Our proposed dynamical system acts on the product space (ℝd x SO(d))N. The evolution of centroids can be described by the coupled linear second-order dynamical system with diffusive linear couplings, whereas rotations for the matching of vertices are described by the Lohe matrix model on SO(d)N. In particular, the Lohe matrix model has been derived from some set of physical principles compared to previous works in which the Lohe matrix model were employed as a system dynamics. This is a contrasted difference between earlier works on the Lohe matrix model which has been adopted a priori for an aggregate modeling of matrices. We also provide an analytical result leading to the complete shape matchings for an ensemble of congruent polytopes, and several numerical examples to illustrate analytical results visually. [ABSTRACT FROM AUTHOR]

المؤلفون: Seung-Yeal Ha¹ syha@snu.ac.kr, Hansol Park² hansol960612@snu.ac.kr

المصدر: SIAM Journal on Applied Dynamical Systems. 2020, Vol. 19 Issue 2, p1312-1342. 28p.

مصطلحات موضوعية: *SPHERES, *INVARIANT sets, *CONSERVED quantity, *TENSOR fields, *AGGREGATION operators

مستخلص: We study emergent behaviors of the Lohe hermitian sphere (LHS) model which is an aggregation model on Cd. The LHS model is a complex analogue of the Lohe sphere model on Rd, and hermitian spheres are invariant sets for the LHS dynamics. For the derivation of the LHS model, we use a top-down approach, namely a reduction from a high-rank aggregation model, the Lohe tensor model. The Lohe tensor model is a first-order aggregation model on the space of tensors with the same rank and sizes, and it was first proposed by the authors in a recent work [J. Stat. Phys., 178 (2020), pp. 1268{1292]. In this work, we study how the LHS model appears as a special case of the Lohe tensor model, and for the proposed model, we provide a cross-ratio-like conserved quantity, a sufficient framework for the complete aggregation, and a uniform p-stability estimate with respect to initial data. [ABSTRACT FROM AUTHOR]

المؤلفون: Seung-Yeal Ha¹ syha@snu.ac.kr, Dohyun Kim² dohyunkim@sungshin.ac.kr, Hansol Park³ hansol_park@sfu.ca

المصدر: SIAM Journal on Applied Dynamical Systems. 2022, Vol. 21 Issue 2, p1166-1208. 43p.

مصطلحات موضوعية: *COMPLEX numbers, *MATRIX multiplications, *TENSOR products

مستخلص: A tensor is a multidimensional array of complex numbers, and the Lohe tensor model is an aggregation model on the space of tensors with the same rank and size. It incorporates previously well-studied aggregation models on the space of low-rank tensors such as the Kuramoto model, Lohe sphere, and matrix models as special cases. Due to its structural complexities in cubic interactions for the Lohe tensor model, explicit construction of solutions with specific structures looks daunting. Recently, we obtained completely separable states by associating rank-1 tensors. In this paper, we further investigate another type of solutions, namely ``quadratically separable states"" consisting of tensor products of matrices and their component rank-2 tensors are solutions to the double matrix model whose emergent dynamics can be studied using the same methodology of the Lohe matrix model. [ABSTRACT FROM AUTHOR]

المؤلفون: Fetecau, Razvan C.¹ fetecau@sfu.ca, Seung-Yeal Ha² hansol960612@snu.ac.kr, Hansol Park¹ syha@snu.ac.kr

المصدر: SIAM Journal on Applied Dynamical Systems. 2022, Vol. 21 Issue 2, p1382-1425. 44p.

مصطلحات موضوعية: *DYNAMICAL systems, *GEOMETRIC shapes, *ROTATIONAL motion, *MATRICES (Mathematics)

مستخلص: We derive an explicit form for the Cucker--Smale (CS) model on the special orthogonal group SOp3q by identifying closed form expressions for geometric quantities such as covariant derivative and parallel transport in exponential coordinates. We study the emergent dynamics of the model by using a Lyapunov functional approach and LaSalle's invariance principle. Specifically, we show that velocity alignment emerges from some admissible class of initial data, under suitable assumptions on the communication weight function. We characterize the \omega -limit set of the dynamical system and identify a dichotomy in the asymptotic behavior of solutions. Several numerical examples are provided to support the analytical results. [ABSTRACT FROM AUTHOR]

المؤلفون: Hyejoo Shin¹, Bong-Kwang Jung¹, Seungwan Ryoo¹, Sooji Hong¹, Heonwoo Jeong¹, Hoo-Gn Jeoung¹, Sunhye Kim², Sun Kim², Min-Jae Kim³, Hansol Park⁴, Eom, Keeseon S.⁴, Kaatano, Godfrey M.⁵, Jong-Yil Chai⁶ cjy@snu.ac.kr, Shin, Hyejoo, Jung, Bong-Kwang, Ryoo, Seungwan, Hong, Sooji, Jeong, Heonwoo, Jeoung, Hoo-Gn, Kim, Sunhye

المصدر: Emerging Infectious Diseases. Nov2022, Vol. 28 Issue 11, p2298-2301. 4p. 1 Color Photograph, 1 Diagram, 2 Charts.

مستخلص: A survey of intestinal helminths targeting 1,440 schoolchildren in 12 primary schools on Kome Island (Lake Victoria), Tanzania, revealed small trematode eggs in 19 children (1.3%), seemingly of a species of Haplorchis or Heterophyes. The eggs were molecularly confirmed to be Haplorchis pumilio on the basis of 18S and 28S rDNA sequences. [ABSTRACT FROM AUTHOR]