# Karlstad Applied Analysis Seminar (KAAS)

## Next seminars:

The seminar series of the academic year has finished. New seminars will take place in the fall of 2020.

## Future seminars:

**When: **August 19, Wednesday, 11:00**What: **Homogenization of boundary value problem on a thin structure**Who: **Aiyappan Srinivasan, TIFR-Centre for Applicable Mathematics, Bangalore India**Where: ***Zoom:* https://kau-se.zoom.us/j/64141214118 (Meeting ID: 641 4121 4118)

**Abstract: **In this talk, we will discuss the homogenization of a boundary value problem posed on a domain with an oscillating boundary. The homogeneous Neumann condition is prescribed on the oscillating part of the boundary. We will explain the homogenization results in the sense of weak $L^2$ convergence of the solutions and their flows. We will look into the application of unfolding operators to various types of oscillating domains.

**When: **August 26, Wednesday, 10:15**What: **On modeling the behavior of pedestrians near walls and the mean-field approach to crowd dynamics**Who: **Alexander Aurell, Princeton University, USA**Where: ***Zoom:* https://kau-se.zoom.us/j/63500991738 (Meeting ID: 635 0099 1738)

**Abstract: **In macroscopic models for the motion of pedestrian crowds, the interaction with walls is often modelled with Neumann-type boundary conditions on the pedestrian density. The interpretation of this type of constraints on a microscopic (individual) level is a pedestrian path reflecting at the walls. Pedestrians however do not reflect on walls, their movement is slowed down by the impact and they need some time to choose a new direction of motion. In the mean-field approach to pedestrian crowd modeling the interaction between pedestrians is assumed to be symmetric and weak, and can be approximated by an interaction with a mean-field (typically a functional of the pedestrian density). One of the main attractions of the mean-field approach is that it connects the macroscopic (pedestrian density) and the microscopic (pedestrian path) description of a crowd. Optimal control in the mean-field approach is well understood for density problems with Neumann conditions. However, for more elaborate mean-field couplings, it is not understood how to control the corresponding paths (reflected SDEs). In this presentation a system of SDEs of mean-field type will be introduced that aims to resolve the two concerns stated above. It lets pedestrians spend some time at the boundary, move along the boundary, and choose new direction of motion when they re-enter the domain. The model is shown to have no strong solution but a unique weak solution and it can be optimally controlled in the weak sense using a Pontryagin’s type maximum principle.

**When: **T.B.A.**What: **Topological and Interfacial Effects on the Glass Transition in Confined Polymers**Who: **Alexey V. Lyulin, Department of Applied Physics, Technische Universiteit Eindhoven, The Netherlands**Where:** T.B.A.

**Abstract: **Glasses in general, and polymer glasses in particular, are not, perhaps surprisingly, technically solid in a crystalized form, but are substances frozen in a liquidlike structure. Many fundamental questions remain as to exactly how glasses form, transitioning from flowing liquid-like state to solid polymer glass. A central factor materials scientists study is the temperature where this occurs, the glass-transition temperature Tg. After some introduction, I will discuss in more detail the recent results of the molecular-dynamics computer simulations of atactic polystyrene (PS), for the bulk and free-standing films, and for both linear and cyclic polymers. Simulated volumetric glass-transition temperatures ([1,2] show a strong dependence on the film thickness below 10 nm [3]. Our studies reveal that the fraction of the chain-end groups is larger in the interfacial layer with its outermost region approx. 1 nm below the surface than it is in the bulk. The enhanced population of the end groups is expected to result in a more mobile interfacial layer and the consequent dependence of Tg on the film thickness. In addition, the simulations show an enrichment of backbone aliphatic carbons and concomitant deficit of phenyl aromatic carbons in the interfacial film layer. This deficit would weaken the strong phenyl-phenyl aromatic interactions and, hence, lead to a lower film-averaged Tg in thin films, as compared to the bulk sample. To investigate the relative importance of the two possible mechanisms (increased chain ends at the surface or weakened p-p interactions in the interfacial region), the data for linear PS are compared with those for cyclic PS. For the cyclic PS the reduction of the glass-transition temperature is also significant in thin films, albeit not as much as for linear PS. Moreover, the deficit of phenyl carbons in the film interface is comparable to that observed for linear PS. Therefore, chain-end effects alone cannot explain the observed pronounced Tg dependence on the thickness of thin PS films; the weakened phenyl-phenyl interactions in the interfacial region seems to be an important cause as well [4]. I will also discuss the interface characteristics of polystyrene in free-standing thin films and on a graphite surface simulated employing an explicit all-atom force field [5]. References [1] Ediger, M.D.; Forrest, J.A. Macromolecules 2014, 47, 471-478. [2] Barrat, J.-L.; Baschnagel, J.; Lyulin, A.V. Soft Matter 2010, 6, 3430-3446. [3] Hudzinskyy, D.; Lyulin, A.V.; Baljon, A.R.C.; Balabaev, N.K.; Michels, M.A.J. Macromolecules 2011, 44, 2299-2310. [4] A. V. Lyulin, N. K. Balabaev, A. R.C. Baljon, G. Mendoza, C. W. Frank and Do Y. Yoon, J. Chem. Phys. 2017, 146, 203314. [5] S. Lee, A. V. Lyulin, C. W. Frank, Do Y. Yoon, Polymer, 2017, 116, 540-548.

**When:** T.B.A.**What**: On robotics, falling cat, parallel parking and stock trading ** ****Who**: Maria Ulan, Linnaeus University**Where:** T.B.A.

**Abstract**: Configuration spaces of many real mechanical systems appear to be manifolds with singularity. A singularity often indicates that geometry of motion may change at the singular point of configuration space. During the talk, I will present one possible solution by considering a certain algebra instead of the configuration space, with a structure completely determined by the geometry of the singularity. We will discussed how one could describe different types of complex motion. We will start with simple examples like linkages, manifolds with corners. Then we will discussed examples with given motion of subsystem. Finally, we will consider some classical reachability problems. At the end of the talk, I will show some applications in industry.

**When:** T.B.A.**What**: Some methods of approximation by spline functions ** ****Who**: Adrian Branga, Lucian Blaga University, Sibiu, Romania**Where:** T.B.A.

**Abstract**: We investigate some classes of spline functions, which can be used to approximate linear functionals defined on unidimensional Sobolev spaces, as well as to find the approximate solutions of some classes of ordinary differential equation and partial derivative equations. The existence and uniqueness of the approximate solution, together with convergence criteria for these methods and estimates of the approximation error are studied and some computational results are given.

**When:** T.B.A.**What**: Ideals generated by quadratic forms in the exterior algebra** ****Who**: Veronica Crispin Quinonez, Uppsala University**Where:** T.B.A.

**Abstract**: (joint work with S. LUNDQVIST and G. NENASHEV) There is a longstanding conjecture due to Fröberg about the minimal Hilbert series of C[x_1, ..., x_n]/(f_1,...,f_r), where f_i are homogeneous forms. We introduce the problem and give some results.

Now let E_n denote the Exterior algebra on n generators over C. It is natural to believe that the Hilbert series of E_n/(f_1,...,f_r) should be equal to the conjectured series in the commutative case, if the f_i's are generic forms of even degree. In 2002, Moreno and Snellman showed it to be true for only one generic form f. However, the same year Fröberg and Löfwall gave a counterexample for the case of two generic forms.

We use the structure theory of pairs of skew-symmetric matrices to study the Hilbert function of two generic quadratic forms f and g in E_n. Further, we use combinatorial methods to describe the Hilbert series of E_n/(f,g). Among our results, we have a conjecture for the minimal Hilbert series.