A method for solving zero-finding problems is developed by tracking homotopy paths, which define connecting channels between an auxiliary problem and the objective problem. Current algorithms' success highly relies on empirical knowledge, due to manually, inherently selected paths. This work introduces a based Theory of Functional Connections (TFC). The TFC-based implicitly defines infinite fro...

Loxodromic arcs are shown to be the maximizers of inversive arclength, which is invariant under Möbius transformations. Previously, these arcs were known to be extremals. The first result says that at any loxodromic arc, the inversive arclength functional is concave with respect to a non-trivial perturbation that fixes the circle elements at the endpoints. The second result says that among curv...

Uniform endpoint Lorentz norm improving estimates for convolution operators with affine arclength measure supported on simple plane curves are established. The estimates hold for a wide class of simple curves, and the condition is stated in terms of averages of the square of the affine arclength weight, extending previously known results. MSC: Primary 44A35; secondary 42B35

In this work we propose a set of tools for the parallel application of pseudo-arclength continuation to a class of systems for which the right hand side can be properly represented by a time numerically calculated evolution operator. For example, the reverse flow reactor and the reactors network with periodically switched inlet and outlet sections belong to this class of system. To conduct a dy...

We study the simplest geometrical particle model associated with null paths in fourdimensional Minkowski space-time. The action is given by the pseudo-arclength of the particle worldline. We show that the reduced classical phase space of this system coincides with that of a massive spinning particle of spin s = α/M , where M is the particle mass, and α is the coupling constant in front of the a...

The numerical simulation of viscoelastic fluid flow becomes more difficult as a physical parameter, the Weissenberg number, increases. Specifically, at a Weissenberg number larger than a critical value, the iterative nonlinear solver fails to converge, a phenomenon known as the High Weissenberg Number Problem. In this work we describe the application and implementation of continuation methods t...

We study the nonlinear dynamics of a tapping mode atomic force microscope with tip-surface interactions that include attractive, repulsive, and capillary force contributions using numerical techniques tailored for hybrid or discontinuous dynamical systems that include forward-time simulation with event handling and numerical pseudo-arclength continuation. We find four branches of periodic solut...

In this paper, we first compute the multiple non-trivial solutions of the Schrödinger equation on a square, by using the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory, combinedwith Legendre pseudospectralmethods. Then, starting from the non-trivial solution branches of the corresponding nonlinear problem, we further obtain the whole positive solution branch with D4 symmetr...

Abstract Let $${\mathcal {A}}$$ A be the class of functions $$\begin{aligned} g(z)=z+\sum _{n=2}^{\infty }a_nz^n \end{aligned}$$ g ( z ) = + <mml:munder...

This study considers a novel nonlinear system, namely cable-stayed beam with tuned mass damper (cable-beam-TMD model), allowing the description of energy transfer among beam, cable and TMD. In this vibration TMD is involved one-to-one-to-one internal resonance modes investigated when external primary occurs. Galerkin’s method utilized to discretize equations motion cable. way, set ordinary diff...