# MAM seminars Spring term 2016

Higher seminars in the subject Mathematics/Applied Mathematics, Spring term 2016. School of Education, Culture and Communication (UKK), Mälardalen University.

*Wednesdays afternoon is the normal time for MAM seminars with deviations when necessary. The program is always provisional. The information about each specific talk at MAM seminar becomes final the day before. Suggestions for talks at MAM seminar are very welcome to Prof. Sergei Silvestrov sergei.silvestrov@mdh.se. *

**P****rogram for Mathematics and Applied Mathematics seminar. **

**January 20, 2016, Wednesday, 15.30-17.00**

Location: U3-083 (Hilbert room), Västerås, UKK, Mälardalen University

**Obs! **Two review presentations on two selected research directions by two MAM PhD students as final part of examination in PhD course Integration and Measure Theory

*Presentation 1*

Speaker: Christopher Engström, Division of Applied Mathematics, Mälardalen University

__Title:__

The Wiener process

__Abstract:__

The Wiener process also known as Brownian motion is a well known and extensively used process in probability theory and its applications. The Wiener process is due to Norbert Wiener and his work on constructing a rigorous mathematical model for Brownian motion (movement of small particles in a fluid), especially in the limiting case as the number of particles increase and the size decrease.

We will start with the classical problem of trying to explain the movement of small particles in a fluid and how the limiting case where we have an infinite number of particles of infinitesimal size gives us the Wiener process. While the Wiener process is primarily applied in probability theory and applications in statistics, our aim is to just as Wiener did, construct the Wiener process from the perspective of real analysis.

*Presentation 2*

Speaker: Jonas Österberg**, **Division of Applied Mathematics, Mälardalen University

__Title:__

The classical moment problems; from moments to measures

__Abstract:__

We study the classical moment problems of Hamburger, Stieltjes and Hausdorff. This class of problems assumes the knowledge of a finite or infinite sequence of real numbers, and asks whether these can be produced by a measure on the real line as the values of integrals of powers of the variable, the moments. The term moment problem was introduced by Stieltjes in 1894-1895 in relation to continued fractions, in the same work that he defined the Stieltjes integral.

We will follow a modern treatment of this class of problems, without continued fractions, and in preparation for generalizations beyond the real line. We will find criteria for existence and uniqueness and will provide expressions for constructing solutions when there are many. We will also look at the truncated moment problem, when only a finite number of moments are known, and discuss it’s relation to orthogonal polynomials.

**January 26, 2016, Tuesday, 10.15-12.00**

Location: U3-083 (Hilbert room), Västerås, UKK, Mälardalen University

Speaker: Abdenacer Makhlouf, University of Haute Alsace, Mulhouse, France

*Expert lecture series, intensive **PhD course *(Lecture 1)

__Title:__

Basics on Algebraic Geometry and Gröbner Basis

__Abstract:__

The aim of the lectures is to provide the basics of algebraic geometry including polynomials, ideals algebraic varieties, Groebner Basis and Nullstelensatz Theorem. Moreover, we show the correspondence between algebraic and geometric objects and perform computations using a computer algebra system in various cases and problems.

**January 27, 2016, Wednesday, 10.15-12.00**

Location: U3-083 (Hilbert room), Västerås, UKK, Mälardalen University

Speaker: Abdenacer Makhlouf, University of Haute Alsace, Mulhouse, France

*Expert lecture series, intensive **PhD course *(Lecture 2)

__Title:__

Basics on Algebraic Geometry and Gröbner Basis

__Abstract:__

The aim of the lectures is to provide the basics of algebraic geometry including polynomials, ideals algebraic varieties, Groebner Basis and Nullstelensatz Theorem. Moreover, we show the correspondence between algebraic and geometric objects and perform computations using a computer algebra system in various cases and problems.

**January 27, 2016, Wednesday, **15.30-16.30

Location: U3-083 (Hilbert room), Västerås, UKK, Mälardalen University

Speaker: Abdenacer Makhlouf, University of Haute Alsace, Mulhouse, France

__Title:__

Recent developments in Hom-algebras and Hom-bialgebras and perspectives

__Abstract:__

In the last years, many concepts and properties from classical algebraic theories have been extended to the framework of Hom-structures. In this talk, we deal with a recent generalization involving two linear maps. We mainly discuss constructions and representations of BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras, as well as related topics.

**January 29, 2016, Friday, 16.00-17.00**

Location: U3-083 (Hilbert room), Västerås, UKK, Mälardalen University

Speaker: Abdenacer Makhlouf, University of Haute Alsace, Mulhouse, France

*Expert lecture series, intensive **PhD course *(Lecture 3)

__Title:__

Basics on Algebraic Geometry and Gröbner Basis

__Abstract:__

The aim of the lectures is to provide the basics of algebraic geometry including polynomials, ideals algebraic varieties, Groebner Basis and Nullstelensatz Theorem. Moreover, we show the correspondence between algebraic and geometric objects and perform computations using a computer algebra system in various cases and problems.

**February 3, 2016, Wednesday, 15.30-16.30**

Location: U3-083 (Hilbert room), Västerås, UKK, Mälardalen University

**OBS! **Talk by PhD student in MAM research environment

Speaker: Alex Behakanira Tumwesigye, Division of Applied Mathematics, Mälardalen University and Department of Mathematics, Makerere University, Kampala, Uganda

__Title:__

On the interplay between one-dimensional dynamical systems and operator algebras

__Abstract:__

In this talk, I discuss some interesting results in the interplay between dynamical systems and operator algebras. I will give an explicit description of the interplay between periodic orbits of one-dimensional piecewise polynomial maps and commutativity of monomials for special operators on a Hilbert space satisfying certain commutation relations. I will also give a description of the maximal commutative subalgebra in a crossed product algebra.

**March 9, 2016, Wednesday, 15.30-16.30**

Location: U3-083 (Hilbert room), Västerås, UKK, Mälardalen University

Speaker: Professor Martin Ostoja-Starzewski, University of Illinois at Urbana-Champaign, USA

__Title:__

Randomness in mechanics of materials

__Abstract:__

Microstructural randomness is present in just about all solid materials. When dominant (macroscopic) length scales are large relative to microscales, one can safely work within classical, deterministic solid mechanics. However, when the separation of scales does not hold, various concepts of continuum mechanics need to be re-examined and new methods developed. In this talk we focus on scaling from a Statistical Volume Element (SVE) to a Representative Volume Element (RVE). Starting from the Hill-Mandel homogeneity condition, without assuming any spatial periodicity, the RVE is approached in terms of two hierarchies of bounds stemming, respectively, from uniform kinematic and static boundary value problems set up on the SVE. This is illustrated in various settings: conductivity, linear or finite (thermo) elasticity, plasticity, viscoelasticity, and Darcy permeability. In the latter case, should the pores’ sizes be comparable to nanoscales, one may have to account for violations of the second law of thermodynamics. The entire methodology can also be extended to homogenization of random media by micropolar (Cosserat) rather than by classical (Cauchy) continua. This methodology also forms a systematic basis for setting up of tensor random fields as input for stochastic PDE and stochastic finite element methods.

**Martin Ostoja-Starzewski** did his undergraduate studies (1977) at the Kraków University of Technology, Poland, followed by Master’s (1980) and Ph.D. (1983) degrees at McGill University, Canada, all in mechanical engineering. His research interests are primarily in (thermo) mechanics of random and fractal media, advanced continuum theories, as well as aerospace, bio- and geo-physical applications. He wrote 170+ journal papers as well as two books: 1. *Microstructural Randomness and Scaling in Mechanics of Materials*, CRC Press (2007); 2. *Thermoelasticity with Finite Wave Speeds*, Oxford University Press (2009). He also (co-) edited 14 books/journal special issues and co-organized various meetings. He is/was on editorial boards of many journals, including *J. Thermal Stresses*, *Probabilistic Engineering Mechanics*, *ASME J. Applied Mechanics*, *Int. J. Damage Mech.*, *Archive of Applied Mechanics*, *Acta Mechanica*, *Mathematics and Mechanics of Complex Systems*, and *Mechanics Research Communications.* He is also co-Editor of the *CRC Modern M**echanics and Mathematics Series* and Fellow of ASME, AAM, WIF, as well as Assoc. Fellow of AIAA. In the winter of 2012 he was Timoshenko Distinguished Visitor at Stanford. Presently, he is Site co-Director of NSF Industry/University Cooperative Research Center for Novel High Voltage/Temperature Materials and Structures.

**April 6, 2016, Wednesday, 15.30-16.30**

Location: U3-083 (Hilbert room), Västerås, UKK, Mälardalen University

Speaker: Nils Muellner, Postdoc, Division of Embedded Systems, School of Inovation, Design and Engineering

__Title:__

Dijkstra (1974), Markov (1906), Cantor (1896): How infinity affects self-stabilization

__Abstract:__

Self-stabilization was introduced by Dijkstra in 1974. Yet, to this day, computer scientists and engineers struggle with its intricacies, leading to more than 20 different variants. One of the challenges is proving that a system converges to a distinguished set of states in a transition model in finitely many computation steps. The talk introduces self-stabilization and shows how systems can be modelled via Markov chains along with examples. Finally, the link to software testing is drawn and the TOCSYC project is presented.

**April 13, 2016, Wednesday, 15.30-16.30**

Location: U3-083 (Hilbert room), Västerås, UKK, Mälardalen University

Speaker:

Marta Leniec, PhD Student, Department of Mathematics, Uppsala University

__Title:__

Pricing and hedging of default-sensitive contingent claims for informed investors

__Abstract:__

We study the problem of pricing default-sensitive contingent claims for an informed investor who observes the stock price process as well as possesses additional information containing the knowledge of the default time from the very beginning. Under the assumption that the underlying default-free market is complete for a regular investor, i.e. an agent who observes only the stock price, and that the defaultable market is arbitrage-free for the informed investor, we show that any default-sensitive contingent claim has a unique price for the informed investor. Moreover, this price can be expressed in terms of prices of default-free contingent claims for the regular investor.

**May 18, 2016, Wednesday, 15.30-16.30**

Location: U3-083 (Hilbert room), Västerås, UKK, Mälardalen University

Speaker:

Magnus Aspenberg, Center for Mathematical Sciences, Lund University

__Title:__

Critically nonrecurrent rational maps

__Abstract:__

We consider critically non-recurrent meromorphic maps on the Riemann sphere. Given a rational map f, we say that it satisfies the Misiurewicz condition if the set of critical points on the Julia set is non-recurrent. A weaker condition is if every critical point c is non-recurrent. These maps are called semi-hyperbolic (Carleson, Jones, Yoccoz). I prove measure theoretic statements about these functions in the parameter space of rational maps of any fixed degree at least 2.

**May 25, 2016, Wednesday, 15.30-16.30 **

Location: U3-083 (Hilbert room), Västerås, UKK, Mälardalen University

Speaker: Prof. Viktor Abramov, Institute of Mathematics, University of Tartu, Estonia

__Title __

Ternary Grassmann algebra, differential calculus over ternary Grassmann algebra and possible generalizations of Dirac operator

__Abstract:__

We consider a ternary analog of Grassmann algebra constructed with the help of cyclic permutations of triple generators and primitive 3rd root of unity. We extend this ternary Grassmann algebra to an algebra with involution. We construct an algebra which includes both a ordinary Grassmann algebra and ternary Grassmann algebra as subalgebras. Making use of noncommutative differential calculus developed by Borowiec, Kharchenko and Oziewicz we develop a calculus of partial derivatives on a ternary Grassmann algebra. We use these derivatives to find a possible generalization of Dirac operator and consider possible applications of this operator in quark model.

**May 26, 2016, Thursday, 13.15-15.30+**

*The public defense of Alex Behakanira Tumwesigye's licentiate thesis in Mathematics and Applied Mathematics*

Location: Kappa, Västerås, UKK, Mälardalen University

Speaker: Alex Behakanira Tumwesigye, Mathematics and Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Sweden and Makerere University, Kampala, Uganda

__Title:__

On one-dimensional dynamical systems and commuting elements in non-commutative algebras

__Abstract:__

*Summary.*

This work is about commutativity which is a very important topic in mathematics, physics, engineering and many other fields. Two processes are said to be commutative if the order of "operation" of these processes does not matter. A typical example of two processes in real life that are not commutative is the process of opening the door and the process of going through the door.

Another example of the importance of commutativity comes from signal processing. Signals pass through filters (often called operators on a Hilbert space by mathematicians) and commutativity of two operators corresponds to having the same result even when filters are interchanged.

Many important relations in mathematics, physics and engineering are represented by operators satisfying a number of commutation relations.

In one part of this thesis I treat commutativity of monomials of pairs of operators satisfying certain commutation relations. This means that the operators do not actually commute but there is an explicit relation between the two possible products of the operators. I consider products of powers of the operators, called monomials, and derive commutativity conditions of the said monomials. I show that this is related to the existence of periodic points of certain one-dimensional dynamical systems. In the second part of the thesis, I treat maximal commutative subalgebras of crossed products of algebras of piecewise constant functions with the group of integers. By the crossed product of an algebra with the group of integers I mean a generalization of Laurent polynomials with coefficients from the algebra and with the algebra multiplication incorporating the dynamical system action of the group of integers. I describe the commutant (set of elements that commute with a given set) and the center (set of elements that commute with the whole algebra) in a number of cases connecting it to the properties of the dynamical system action on the algebra of piecewise constant functions.

**June , 2016, Wednesday, 15.30-16.30**

Location: U3-083 (Hilbert room), Västerås, UKK, Mälardalen University

Speaker:

__Title:__

__Abstract:__