Piga, Sanhueza-Matamala, and Schacht recently established that the codegree Tur & aacute;n density of 3-uniform tight cycles C & ell; is 1/3 for & ell;is an element of{10,13,16} and for all & ell;>= 19 . In this note, we extend their proof to determine the codegree Tur & aacute;n density of the 3-uniform tight cycle C11 , thereby completing the picture for tight cycles of length at least 10.
Computing the intersection curve of two surfaces is fundamental in CAD and geometric modeling but remains challenging because of degeneracy such as tangential intersection and the difficulty of recovering correct topology. Robustness is therefore essential for a reliable surface–surface intersection module. In this paper, we present a robust algorithm for intersecting two low-degree (less than or equal to four) algebraic surfaces common in geometric modeling (e.g., quadrics, toric surfaces, swept surfaces, and surfaces of revolution). The method projects the spatial intersection onto a plane and computes singular and critical points of the resulting planar algebraic curve using volume functions associated with the curve. Crucially, the exact number of singular and critical points is determined a priori, which guarantees correct planar topology in the subsequent tracing stage. The planar curve is traced starting from singular, boundary, and critical points until all components are recovered, and the traced components are lifted back to 3D using subresultant techniques to produce the spatial intersection. We verify our method using several examples, which contain degenerate cases such as tangency, and the results demonstrate the robustness of our algorithm. The proposed algorithm has the potential to solve the stability problem in low-degree algebraic surface intersection in geometric kernel.
This paper investigates the long-time behavior of a nonlinear wave equation with a time-varying delay and an external forcing term on a bounded domain. Under suitable assumptions on the nonlinear term, the external force, and the delay function, the existence and uniqueness of a global weak solution are recalled within the appropriate energy framework. An adapted Lyapunov functional incorporating the delay history is then constructed. By verifying the gradient property, the boundedness of the Lyapunov sublevel sets and stationary solutions, and the asymptotic smoothness of the associated dynamical system, we establish the existence of a compact global attractor. Furthermore, when the external force vanishes, an appropriate perturbed energy functional is employed to prove the exponential decay of the energy under a suitable relation between the damping coefficient, the delayed feedback coefficient, and the derivative of the delay function.
A strongly balanced constrained clustering (SBCC) problem requires that clusters satisfy the equality constraints on cluster weights while it is also required to ensure connectivity of each cluster in practical scenarios. However, existing methods usually transform the strong balancing constraints into inequality constraints and obtain an approximate solution to the SBCC problem, resulting in the failure to guarantee the accuracy of the solution and cluster’s connectivity. In this paper, we propose a balance adjustment method, by combining a shelved-retrieved strategy and an adjacent allocation method to get the high quality solution satisfying the connectivity of each cluster. To verify the validity and rationality of the proposed method, we conducted experiments on three synthetic datasets and in two real-world scenarios. The results show that our method can generate the clusters fulfilling the constraints in the SBCC, which demonstrates that our method outperforms existing methods. Therefore, the balance adjustment method is capable of solving the SBCC problem and provides high quality solutions, which allows for on-demand resource allocation.
In this paper, the vertical rotation interval of a homeomorphism f:𝕋^2→𝕋^2 homotopic to Dehn twist is considered. We first prove that if the vertical rotation interval of f has an irrational endpoint, then we can find a homeomorphism g arbitrarily C^0 -close to f such that ρ _V(G)∩ρ _V(F)^c∅ . It is also proved that the vertical displacement of points under n iterates of a lift of an area-preserving C^1+ϵ -diffeomorphism f has bounded drift away from nρ at any ρ∈∂ρ _V(F) when ρ _V(F) has interior. We also investigate the vertical rotation sets of minimal sets, which are proved to be closed intervals (possibly non-degenerate). Further, we show that if f has bounded mean vertical-motion on a minimal set ℳ , then f|_ℳ is regularly semi-conjugate to the irrational rotation on 𝕋^1 .
