# Karlstad Applied Analysis Seminar (KAAS)

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## Future seminars:

**When:** 10:30AM-11:30AM, 22 March 2023**What: ** Constitutive models for rubber**Who:** Patrick Schneider, Department of Simulation Methods and Continuum Mechanics,German Institute for Rubber Technology e. V. (DIK e.V.), Germany.**Where: **https://kau-se.zoom.us/j/63372480848

**Abstract: **Industrial rubber consists of crosslinked polymers usually reinforced by active filler particles such as carbon black. In the first part of the talk, we will discuss the complex material behaviour of rubber and give a comprehensive overview about general ideas and methods to model its constitutive behaviour within a continuum mechanical setting. In the second part of the talk, a short overview will be given about recent research activities of the department of simulation and continuum mechanics of the DIK.** **

**When:** 10:30AM-11:30AM, 5 April 2023**What: ** End-Point Norm Estimates for Cesàro and Copson Operator**Who:** Bizuneh Minda Demissie,Department of Mathematics and Computer Science, Karlstad university, Sweden and Department of Mathematics, Addis Ababa university, Ethiopia**Where: ** Offline: 9D321, Online: https://kau-se.zoom.us/j/63372480848

**Abstract: **For a large class of operators acting between weighted l^∞ spaces, exact formulas are given for their norms and the norms of their restrictions to the cones of non-negative sequences; non- negative, non-increasing sequences; and non-negative, non-decreasing sequences. The weights involved are arbitrary nonnegative sequences and may differ in the domain and codomain spaces. The results are applied to the Cesáro and Copson operators, giving their norms and their distances to the identity operator on the whole space and on the cones. Simplifications of these formulas are derived in the case of these operators acting on power-weighted l^∞. As an application, best constants are given for inequalities relating the weighted l^∞ norms of the Cesáro and Copson operators both for general weights and for power weights.

**When:** 11:00AM-12:00AM, 26 April 2023**What: ** Freeform Reflector Design with a Scattering Surface**Who:** Vì Kronberg, Eindhoven University of Technology, Netherlands**Where: ** Offline: 21A345, Online: https://kau-se.zoom.us/j/63372480848

**Joint KAAS seminar & Research Seminars in Physics**

**Abstract:**

In this talk, we shall derive a model of surface light scattering suitable for use when solving the inverse problem of reflector design within the field of computational illumination optics, i.e., the problem of finding a freeform reflector (curved mirror) such that a given source light distribution is transformed into a specified target light distribution. This problem has been extensively studied for the case of specular reflectors (perfect mirrors) but literature unifying freeform reflectors and surface scattering effects is exceedingly rare.

The scattering model relies on concepts from optimal transportation theory, and can be formulated by endowing a smooth curve/surface (representing the reflector) with so-called microfacet surface roughness. The microfacets are small, tilted, specular (mirror-like) sections superimposed on the macroscopic reflector. By prescribing their orientations along the reflector, a model of surface roughness may be formulated.

In two dimensions, which will be the main focus of this talk, the final expression for the scattered light is a convolution integral, whilst the equivalent expression in three dimensions takes the shape of a Fredholm integral of the first kind. After deriving the convolution integral, we shall show how to go from the theoretical model to the practical application by computing a few reflector surfaces with varying amounts of scattering.