Search results for "ta111"

Ledrappier-Young formula and exact dimensionality of self-affine measures

2017

In this paper, we solve the long standing open problem on exact dimensionality of self-affine measures on the plane. We show that every self-affine measure on the plane is exact dimensional regardless of the choice of the defining iterated function system. In higher dimensions, under certain assumptions, we prove that self-affine and quasi self-affine measures are exact dimensional. In both cases, the measures satisfy the Ledrappier-Young formula. peerReviewed

Mappings of finite distortion between metric measure spaces

2015

We establish the basic analytic properties of mappings of finite distortion between proper Ahlfors regular metric measure spaces that support a ( 1 , 1 ) (1,1) -Poincaré inequality. As applications, we prove that under certain integrability assumption for the distortion function, the branch set of a mapping of finite distortion between generalized n n -manifolds of type A A has zero Hausdorff n n -measure.

Kids Inspire Kids for STEAM

2017

Abstract The goal of the Kids Inspiring Kids in STEAM (KIKS) project was to raise students' awareness towards the multi- and transdisciplinary connections between the STEAM subjects (Science, Technology, Engineering, Arts & Mathematics), and make the learning about topics and phenomena from these fields more enjoyable. In order to achieve these goals, KIKS project has popularized the STEAM-concept by projects based on the students inspiring other students-approach and by utilizing new technologies, tools, open educational resources, and everyday items and materials. Through the students-inspiring-other-students-approach, we have aimed to get participating students developing STEAM activitie…

Gradient walks and $p$-harmonic functions

2017

Nonlinear Liouville Problems in a Quarter Plane

2016

We answer affirmatively the open problem proposed by Cabr\'e and Tan in their paper "Positive solutions of nonlinear problems involving the square root of the Laplacian" (see Adv. Math. {\bf 224} (2010), no. 5, 2052-2093).

Variational parabolic capacity

2015

We establish a variational parabolic capacity in a context of degenerate parabolic equations of $p$-Laplace type, and show that this capacity is equivalent to the nonlinear parabolic capacity. As an application, we estimate the capacities of several explicit sets.

Modelling Environmental Problem-Solving Through STEAM Activities : 4Dframe's Warka Water Workshop

2016

Mappings of Lp-integrable distortion: regularity of the inverse

2016

Let be an open set in ℝn and suppose that is a Sobolev homeomorphism. We study the regularity of f–1 under the Lp-integrability assumption on the distortion function Kf. First, if is the unit ball and p > n – 1, then the optimal local modulus of continuity of f–1 is attained by a radially symmetric mapping. We show that this is not the case when p ⩽ n – 1 and n ⩾ 3, and answer a question raised by S. Hencl and P. Koskela. Second, we obtain the optimal integrability results for ∣Df–1∣ in terms of the Lp-integrability assumptions of Kf.

Acoustic wave guides as infinite-dimensional dynamical systems

2015

We prove the unique solvability, passivity/conservativity and some regularity results of two mathematical models for acoustic wave propagation in curved, variable diameter tubular structures of finite length. The first of the models is the generalised Webster's model that includes dissipation and curvature of the 1D waveguide. The second model is the scattering passive, boundary controlled wave equation on 3D waveguides. The two models are treated in an unified fashion so that the results on the wave equation reduce to the corresponding results of approximating Webster's model at the limit of vanishing waveguide intersection.

Shape identification in inverse medium scattering problems with a single far-field pattern

2016

Consider time-harmonic acoustic scattering from a bounded penetrable obstacle $D\subset {\mathbb R}^N$ embedded in a homogeneous background medium. The index of refraction characterizing the material inside $D$ is supposed to be Holder continuous near the corners. If $D\subset {\mathbb R}^2$ is a convex polygon, we prove that its shape and location can be uniquely determined by the far-field pattern incited by a single incident wave at a fixed frequency. In dimensions $N \geq 3$, the uniqueness applies to penetrable scatterers of rectangular type with additional assumptions on the smoothness of the contrast. Our arguments are motivated by recent studies on the absence of nonscattering waven…