The mathematics of coordinated control of prosthetic arms and manipulators.

Proceedings of the first Applied Solid Mechanics Conference, Glasgow, UK, 26-27 March 1985.

Applied Solid Mechanics

We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object has a non-obvious non-linear geometric interpretation. We recall that the binormal flow is a standard model for the evolution of vortex filaments. We prove the existence of solutions of the binormal flow with smooth trajectories that are as close as desired to curves with a multifractal behavior. Finally, we show that this behavior falls within the multifractal formalism of Frisch and Parisi, which is conjectured to govern turbulent fluids.

Riemann's non-differentiable function and the binormal curvature flow

We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence, this analytical object has a non-obvious nonlinear geometric interpretation. We recall that the binormal flow is a standard model for the evolution of vortex filaments. We prove the existence of solutions of the binormal flow with smooth trajectories that are as close as desired to curves with a multifractal behavior. Finally, we show that this behavior falls within the multifractal formalism of Frisch and Parisi, which is conjectured to govern turbulent fluids. 

/pdf/riemanns-non-differentiable-function-and-the-binormal-kn7nv2io.pdf

Riemann’s Non-differentiable Function and the Binormal Curvature Flow

We consider a fractal refinement of the Carleson problem for pointwise convergence of solutions to the periodic Schrodinger equation to their initial datum. For $\alpha \in (0,d]$ and $s \frac{d}{2(d+2)}\left( d+2-\alpha \right)$ is sufficient for the solution corresponding to every datum in $H^s(\mathbb T^d)$ to converge $\alpha$-almost everywhere.

Convergence over fractals for the periodic Schr\"odinger equation

In this paper, an approximate method combining the finite difference and collocation methods is studied to solve the generalized fractional diffusion equation (GFDE). The convergence and stability analysis of the presented method are also established in detail. To ensure the effectiveness and the accuracy of the proposed method, test examples with different scale and weight functions are considered, and the obtained numerical results are compared with the existing methods in the literature. It is observed that the proposed approach works very well with the generalized fractional derivatives (GFDs), as the presence of scale and weight functions in a generalized fractional derivative (GFD) cause difficulty for its discretization and further analysis.

/pdf/finite-difference-collocation-method-for-the-generalized-15hfcr03.pdf

Finite Difference–Collocation Method for the Generalized Fractional Diffusion Equation

In this paper, we consider the evolution of the Vortex Filament equation (VFE): \begin{equation*} \mathbf X_t = \mathbf Xs \wedge \mathbf Xss, \end{equation*} taking $M$-sided regular polygons with nonzero torsion as initial data. Using algebraic techniques, backed by numerical simulations, we show that the solutions are polygons at rational times, as in the zero-torsion case. However, unlike in that case, the evolution is not periodic in time; moreover, the multifractal trajectory of the point $\mathbf X(0,t)$ is not planar, and appears to be a helix for large times. 
These new solutions of VFE can be used to illustrate numerically that the smooth solutions of VFE given by helices and straight lines share the same instability as the one already established for circles. This is accomplished by showing the existence of variants of the so-called Riemann's non-differentiable function that are as close to smooth curves as desired, when measured in the right topology. This topology is motivated by some recent results on the well-posedness of VFE, which prove that the selfsimilar solutions of VFE have finite renormalized energy.

On the Evolution of the Vortex Filament Equation for regular $M$-polygons with nonzero torsion

In this work, we propose a mathematical model for a physical problem based on the movement of a metal piece held by a robot. Using the principles of Kirchoff plate theory, a set of equations determining stresses and deformations caused during the motion, have been provided. We also discuss possible numerical treatment of these equations and finally, a solution to the one-dimensional analog of the problem has been presented.

Safe trajectory of a piece moved by a robot.

Uno de los fenomenos mas interesantes en la literatura sobre fluidos es la aparicion yevolucion de filamentos de vortice. Algunos de sus ejemplos en el mundo real son los anillosde humo, los remolinos y los tornados. En el caso de un fluido ideal, se han propuesto variosmodelos y ecuaciones para describir esta evolucion, de entre los cuales la VFE (vortexfilament equation) ha ganado protagonismo recientemente debido a su simplicidad ypropiedades geometricas. La ecuacion aparecio por primera vez en el trabajo de Da Rios aprincipios del siglo XX como una aproximacion de la dinamica de un filamento de vortice. Estemodelo generalmente se conoce como LIA (localized induction approximation). En estetrabajo, examinamos la evolucion de la VFE para curvas poligonales regulares desde un puntode vista numerico y teorico, tanto en la geometria euclidiana como en la hiperbolica.En la primera parte de la tesis, observamos la evolucion de la VFE tomando poligonosregulares de M lados con torsion no nula como datos iniciales en el espacio euclidiano.Usando tecnicas algebraicas, respaldadas por simulaciones numericas, mostramos que lassoluciones son poligonos en tiempos racionales, como en el caso de torsion nula. Sinembargo, a diferencia de ese caso, la evolucion no es periodica en el tiempo, y ademas, latrayectoria multifractal de un punto no es plana y parece ser una helice para tiempos grandes.Estas nuevas soluciones de la VFE pueden usarse para ilustrar numericamente que sussoluciones regulares proporcionadas por helices y lineas rectas comparten la mismainestabilidad que la establecida para los circulos. Esto se logra mostrando la existencia devariantes de la funcion no diferenciable de Riemann que estan tan cerca de las curvas suavescomo se desee, cuando se mide en la topologia correcta. Esta topologia esta motivada poralgunos resultados recientes que muestran que la VFE esta bien definida y que sus solucionestienen energia renormalizada finita.En el resto del trabajo, profundizamos en la configuracion hiperbolica y examinamos laevolucion de la VFE para un l-poligono plano, es decir, un poligono plano regular en el espaciode Minkowski. A diferencia del caso euclidiano, un l-poligono plano es abierto, lo que hace queel problema sea mas complicado desde un punto de vista numerico. Despues de probar variosmetodos numericos, concluimos que una discretizacion del espacio por diferencias finitascombinada con un metodo explicito de Runge--Kutta en tiempo brinda los mejores resultadosnumericos tanto en terminos de eficiencia como de precision. Por otro lado, utilizandoargumentos teoricos, recuperamos la evolucion algebraicamente y, por lo tanto, mostramosque los dos enfoques concuerdan. Durante la evolucion numerica, se ha observado que latrayectoria de una esquina es multifractal, y que a medida que el parametro l tiende a cero,converge a la funcion no diferenciable de Riemann.Ademas, como en el caso euclidiano, proporcionamos pruebas numericas solidas parademostrar que en tiempos infinitesimales, la evolucion de la VFE para un l-poligono planocomo dato inicial puede describirse como una superposicion de varios datos iniciales de unaesquina. Como consecuencia, no solo podemos calcular la velocidad del centro de masa del lpoligonoplano teoricamente, sino que la relacion tambien revela propiedades importantes dela trayectoria de sus esquinas que comparamos con su equivalente en el caso euclidiano.Finalmente, una torsion no nula en el caso hiperbolico produce dos tipos diferentes de curvaspoligonales helicoidales, que tambien podemos abordar con las tecnicas numericas y teoricasdesarrolladas hasta ahora. Esto forma parte del trabajo futuro de la tesis.

Vortex Filament Equation for some Regular Polygonal Curves

High Performance Computing for Modelling of Stereolithography Process. (Hochleistungsrechnen für die Modellierung des Stereolithografieprozesses / Calcul à haute performance pour la modélisation du processus de stéréolithographie)

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