# MAM Seminars Autumn term 2016

Higher seminars in the subject Mathematics/Applied Mathematics, Autumn term 2016.

School of Education, Culture and Communication (UKK), Mälardalen University.

**Program for Mathematics and Applied Mathematics seminar. **

**Autumn term 2016**

*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.*

**October 5, 2016, Wednesday, 15.30-16.30**

Location: U2-145, Västerås, UKK, Mälardalen University

Speaker: Anatoliy Malyarenko, Division of Applied Mathematics, MAM, UKK, Mälardalen University

__Title:__

A random field formulation of constitutive laws of continuum physics in all symmetry classes

__Abstract:__

Constitutive laws of continuum physics are expressed in terms of tensor relations between the gradients of primary variables and their fluxes. A general tensor relation can be divided into classes according to its symmetry properties. It is possible to attach one or more material symmetry groups to the fixed point set of each symmetry class. At microscopic length scales, spatial randomness of the material needs to be taken into account in the form of a homogeneous random field. When the body is rotating by any of the attached groups, the field is changing according to the prescribed orthogonal representation of the group acting in the fixed point set. We call such a field isotropic.

We discuss the general form of the spectral expansion for such a field and consider the links between the theory of homogeneous and isotropic random fields and other areas of mathematics. These includes well-known links to Special Functions and Classical Invariant Theory as well as a link to the theory of finite-dimensional convex compacta recently discovered by the authors.

**October 18, October 20, 2016, Wednesday, 15.30-16.30**

*MAM intensive expert lectures series by Professor Abdenacer Makhlouf from UHA France.*

Location: U2-016 (October 18), R2-605 (Ocotber 20), Västerås, UKK, Mälardalen University

Speaker: Professor Abdenacer Makhlouf, Laboratoire de Mathématiques, Informatique et Applications, Université de Haute-Alsace - UHA

http://www.math.uha.fr/makhlouf/

__Title:__

Introduction to Algebraic Geometry and Groebner Basis

__Abstract:__

In the focused course of lectures, I will provide the basics of algebraic geometry, including ideals of polynomials, algebraic varieties and their relationships. Moreover, Groebner basis will be described with several applications.

**October 19, 2016, Wednesday, 15.30-16.30**

Location: U2-145, Västerås, UKK, Mälardalen University

Speaker: Abdenacer Makhlouf, Laboratoire de Mathématiques, Informatique et Applications, University of Haute Alsace, Mulhouse, France

http://www.math.uha.fr/makhlouf/

__Title:__

n-Ary algebras: from Physics to Mathematics

__Abstract:__

Lie algebras and Poisson algebras have played an extremely important role in mathematics and physics for a long time. Their generalizations, known as n-Lie algebras and “Nambu algebras” also arise naturally in physics in many different contexts. For instance, ternary algebras can be used to construct solutions of the Yang-Baxter equation which appeared first in statistical mechanics. Nambu mechanics involves an n-ary bracket and provide a generalization of Hamiltonian mechanics by considering more than one Hamiltonian. The structure of n-ary multiplication also appears in several cases within string theory and noncommutative geometry.

An n-ary algebra is a vector space provided with a multiplication given by an n-linear map.

In this talk, I will review some basics on n-ary algebras, present some key constructions and generalizations. Moreover, I will discuss representation theory and cohomology theory.

**October 26, 2016, Wednesday, 15.30-16.30**

Location: U2-145, Västerås, UKK, Mälardalen University

Speaker: Johan Öinert, Blekinge Institute of Technology, Sweden

__Title:__

Some recent results on graded/filtered non-associative and non-commutative algebra

__Abstract:__

This talk is based on three recent preprints and will focus on three different topics coming from non-commutative and non-associative algebra.

1. Recall that a ring is said to be right (left) quasi-duo if every maximal right (left) ideal of it is two-sided. We will give a characterization of (left and right) quasi-duo differential polynomial rings. (Joint work with Mai Hoang Bien.)

2. For group graded rings we will introduce the notion of a G-controlled ring. For strongly graded rings, it can be seen as a strong form of simplicity. We will give a characterization of G-controlled rings and present some open problems.

3. We will give a complete characterization of simple non-associative unital rings which are graded by hypercentral groups. This result will be applied to dynamical systems and to Cayley-Dickson doublings. (Joint work with Patrik Nystedt.)

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

Location: U2-145, Västerås, UKK, Mälardalen University

Speaker: Professor Nikolai Leonenko, Cardiff University, United Kingdom

__Title:__

Fractional Skellam Processes

__Abstract:__

Recent literature on high frequency financial data includes models that use the difference of two Poisson processes, and incorporate a Skellam distribution for forward prices. The exponential distribution of inter-arrival times in these models is not always supported by data. Fractional generalization of Poisson process, or fractional Poisson process, overcomes this limitation and has Mittag-Lefler distribution of inter-arrival times. We introduce and study a fractional Skellam processes as the differences of two fractional Poisson processes.

(Joint work with A. Kerss and A. Sikorskii).

**November 16****, 2016, Wednesday, 15.15-17.30**

**China-Mälardalen MAM focused international workshop**

**“Non-commutative and non-associative algebra and applications”**

Organiser: Professor Sergei Silvestrov, MAM, Division of Applied Mathematics, UKK, Mälardalen University

Place: U2-145, Mälardalen University, Västerås,

Time: 15.15-17.30, November 16, 2016

Workshop aims to establish joint projects of interest and scientific cooperation between Mathematics and Applied Mathematics (MAM) research environment at Mälardalen University and Chern Institute of Mathematics at Nankai University in Tianjin, China.

**Program**

**15.15-15.55** Professor Chengming Bai, Chern Institute of Mathematics (CIM), Nankai University, Tianjin, China

A brief introduction to pre-Lie algebras

Abstract: In this talk, I will give a brief introduction to pre-Lie algebras, with emphasizing their relations with some related structures. In particular, some constructions and examples of pre-Lie algebras are given.

**16.00-16.40** Dr. Johan Richter, Division of Applied Mathematics, MAM, UKK, Mälardalen University

Simplicity of Associative and Non-associative Ore Extensions

Abstract: I will review the definition of Ore extensions, a non-commutative (associative) generalization of polynomial rings. I will present some results on when Ore extensions are simple. I will then discuss non-associative Ore extensions, which is a new concept, and present some theorems on when they are simple, generalizing the results for associative Ore extensions. The part about non-associative Ore extensions is based on joint work by Patrik Nystedt, Johan Öinert and myself.

**16.45-17.25** Dr. Lars Hellström, Division of Applied Mathematics, MAM, UKK, Mälardalen University

The Network Rewriting Formalism for Expressions in Bi- and Hopf Algebras (and the like)

Abstract: The analysis and transformation of mathematical formulae, particularly expressions, is a core mathematical activity that gave rise to the discipline of algebra; however in recent decades algebra has advanced at a pace that has left the formula language lagging behind. A particular difficulty is seen for algebraic structures (such as bi- and Hopf algebras) that mix operations and co-operations, since these do not fit into the syntactic tree paradigm that is built into the traditional formula language. The way to overcome that is to allow more general graphs as structures of expression formulae, a route pioneered by Penrose with his diagrammatic notation for tensor expressions. Characteristic of this generalisation is that expressions evolve from text-like formulae into being outright drawings.

The network formalism provides a stringent foundation for doing universal algebra with such expression drawings; one ends up working with the PROP defined by generators and relations. Just like an operad can be used to encode the relations defining a certain variety of algebras with operations, a PROP can be used to encode the relations defining a certain variety of algebras with operations and co-operations, such as bialgebras, Hopf algebras, or infinitesimal bialgebras. Network rewriting is a procedural method of exploring the full theory of some algebraic structure given only its defining axioms. In the case of Hopf algebras, the speaker has been able to compute a complete set of rewrite rules.

**November 23****, 2016, Wednesday, 15.30-16.30**

Location: U2-129, Västerås, UKK, Mälardalen University

Speaker: Mårten Gulliksson and Magnus Ögren, Department of Mathematics, School of Science and Technology, Örebro university

__Title:__

Computational Mathematics at Örebro University

__Abstract:__

We will give some examples of problems in computational mathematics that we have worked with during the last 3 years. Among these problems are inverse problems in PDE, homogenization, Chromatography, robot assisted gas detection and Magnetoencephalography. We will also briefly describe a method using damped second order dynamical systems to solve equations and our work on solving PDE using random walks.

**November 30****, 2016, Wednesday, 15.30-16.30**

Location: U2-145, Västerås, UKK, Mälardalen University

Speaker: Joakim Arnlind, Department of Mathematics (MAI), Linköping University

__Title:__

Curvature of Noncommutative Spheres

__Abstract:__

I will give a brief introduction to the concept of curvature in noncommutative geometry as well as explain our recent work on Levi-Civita connections and the Chern-Gauss-Bonnet theorem for noncommutative spheres.

**December 7****, 2016, Wednesday, 15.30-16.30**

PhD defence, MAM, UKK, MDH

Location: U2-016, Västerås, UKK, Mälardalen University

Speaker: Betuel Canhanga, Division of Applied Mathematics, MAM, UKK, Mälardalen University

__Title:__

Asymptotic methods for pricing European options in a market model with two stochastic volatilities

__Abstract:__

Modern financial engineering is a part of applied mathematics that studies market models. Each model is characterized by several parameters. Some of them are familiar to a wide audience, for example, the price of a risky security, or the risk free interest rate. Other parameters are less known, for example, the volatility of the security. This parameter determines the rate of change of security prices and is determined by several factors. For example, during the periods of stable economic growth the prices are changing slowly, and the volatility is small. During the crisis periods, the volatility significantly increases. Classical market models, in particular, the celebrated Nobel Prize awarded Black–Scholes–Merton model (1973), suppose that the volatility remains constant during the lifetime of a financial instrument. Nowadays, in most cases, this assumption cannot adequately describe reality. We consider a model where both the security price and the volatility are described by random functions of time, or stochastic processes. Moreover, the volatility process is modelled as a sum of two independent stochastic processes. Both of them are mean reverting in the sense that they randomly oscillate around their average values and never escape neither to very small nor to very big values. One is changing slowly and describes low frequency, for example, seasonal effects, another is changing fast and describes various high frequency effects. We formulate the model in the form of a system of a special kind of equations called stochastic differential equations. Our system includes three stochastic processes, four independent factors, and depends on two small parameters. We calculate the price of a particular financial instrument called European call option. This financial contract gives its holder the right (but not the obligation) to buy a predefined number of units of the risky security on a predefined date and pay a predefined price. To solve this problem, we use the classical result of Feynman (1948) and Kac (1949). The price of the instrument is the solution to another kind of problem called boundary value problem for a partial differential equation. The resulting equation cannot be solved analytically. Instead we represent the solution in the form of an expansion in the integer and half-integer powers of the two small parameters mentioned above. We calculate the coefficients of the expansion up to the second order, find their financial sense, perform numerical studies, and validate our results by comparing them to known verified models from the literature. The results of our investigation can be used by both financial institutions and individual investors for optimization of their incomes.

**December 8****, 2016, Wednesday, 15.30-16.30**

PhD defence, MAM, UKK, MDH

Location: Kappa, Västerås, UKK, Mälardalen University

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

__Title:__

PageRank in Evolving Networks and Applications of Graphs in Natural Language Processing and Biology

__Abstract:__

This thesis is dedicated to the use of graph based methods applied to ranking problems on the Web-graph and applications in natural language processing and biology.

Chapter 2-4 of this thesis is about PageRank and its use in the ranking of home pages on the Internet for use in search engines. PageRank is based on the assumption that a web page should be high ranked if it is linked to by many other pages and/or by other important pages. This is modelled as the stationary distribution of a random walk on the Web-graph.

Due to the large size and quick growth of the Internet it is important to be able to calculate this ranking very efficiently. One of the main topics of this thesis is how this can be made more efficiently, mainly by considering specific types of subgraphs and how PageRank can be calculated or updated for those type of graph structures. In particular we will consider the graph partitioned into strongly connected components and how this partitioning can be utilized.

Chapter 5-7 is dedicated to graph based methods and their application to problems in Natural language processing. Specifically given a collection of texts (corpus) we will compare different clustering methods applied to Pharmacovigilance terms (5), graph based models for the identification of semantic relations between biomedical words (6) and modifications of CValue for the annotation of terms in a corpus.

In Chapter 8-9 we look at biological networks and the application of graph centrality measures for the identification of cancer genes. Specifically in (8) we give a review over different centrality measures and their application to finding cancer genes in biological networks and in (9) we look at how well the centrality of vertices in the true network is preserved in networks generated from experimental data.

**December 12**** 2016, Wednesday, 15.30-16.30**

Licentiat defence, MAM, UKK, MDH

Location: U2-016, Västerås, UKK, Mälardalen University

Speaker: Xiaomin Qi, Division of Applied Mathematics, MAM, UKK, Mälardalen University

__Title:__

Fixed points, fractals, iterated function systems and generalized support vector machines

__Abstract:__

In this thesis, fixed point theory is used to construct a fractal type sets and to solve data classification problem. Fixed point method, which is a beautiful mixture of analysis, topology, and geometry has been revealed as a very powerful and important tool in the study of nonlinear phenomena. The existence of fixed points is therefore of paramount importance in several areas of mathematics and other sciences. In particular, fixed points techniques have been applied in such diverse fields as biology, chemistry, economics, engineering, game theory and physics. In Chapter 2 of this thesis it is demonstrated how to define and construct a fractal type sets with the help of iterations of a finite family of generalized F-contraction mappings, a class of mappings more general than contraction mappings, defined in the context of b-metric space. This leads to a variety of results for iterated function system satisfying a different set of contractive conditions. The results unify, generalize and extend various results in the existing literature. In Chapter 3, the theory of support vector machine for linear and nonlinear classification of data and the notion of generalized support vector machine is considered. In the thesis it is also shown that the problem of generalized support vector machine can be considered in the framework of generalized variation inequalities and results on the existence of solutions are established.

**Forthcomming in Spring term 2017: **

**January or February, 2016, Wednesday, 15.30-16.30**

Location: U2-129, Västerås, UKK, Mälardalen University

Speaker: Lars Hellström, Division of Applied Mathematics, MAM, UKK, Mälardalen University

__Title:__

How Eigenvalues Solve Polynomial Equation Systems

__Abstract:__

Solving polynomial equation systems is fundamentally a problem in algebraic geometry, and as such striding the algebra--geometry dualism of this subject. Classical methods tend to lean heavily on the algebra side, treating equation solving primarily as a matter of finding better equations for describing the solution set; Gaussian elimination is definitely a matter of this, and there is a tradition of using Gröbner bases like that as well. Elimination orders do however often lead to (more) computationally expensive Gröbner basis calculations, which can make them impractical.

An alternative approach, which turns out to make readily available the coordinates of solution points, is to instead focus on the quotient algebra. On the one hand, any Gröbner basis can be utilised to perform effective calculations in the quotient. On the other hand, the quotient algebra has a geometric interpretation as the algebra of functions on the solution variety, which can be used to for example locate the points of that variety. Numerical approximations can be found through eigenvalue/eigenvector calculations, the practical stability and implementation efficiency of which has been thoroughly studied in recent decades.