# MAM Seminars Autumn term 2018

Higher seminars in the subject Mathematics/Applied Mathematics, Autumn term 2018.

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

**Program for Mathematics and Applied Mathematics seminar (MAM seminar) **

**Autumn term 2018**

*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.*

**September 12, 2018, Wednesday, 15.30-16.30**

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

__Speaker__: Markku Jääskeläinen, Senior Lecturer in Physics, UKK, Mälardalen University

__Title:__ Exploring many-body dynamics of Ultracold gases in confined quantum wells and the quantum physics of self-gravitating microspheres.

__Abstract:__

In this talk I briefly sketch two of the topics I have worked on since finishing my PhD at the Royal Institute of Technology.

The first line of research was largely motivated by a lack of understanding for experimental results at Stanford University on Bose-Einstein condensates of rubidium atoms in optical lattice traps. Simplified spherical-cow type modelling and numerical simulations turned out to be successful, eventually even convincing the experimentalists.

My research in the second area grew out of my attempts of coming to terms with what reality, if any, quantum mechanics actually describes and how gravity at the classical level fits into the picture. Central here are investigations of how non-relativistic gravity influences quantum phenomena of small solid objects. The subject has been attracting increasing interest in recent years and some basic foundational questions remain open. My efforts go inte tackling these with mathematical rather than philosophical tools in order to offer the experimental community something to observe or refute.

**September 26, 2018, Wednesday, 15.30-16.30**

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

__Speaker__: Rafael Reno S. Cantuba, De La Salle University, Manila

__Title:__ Undeformed commutators in the q-deformed Heisenberg algebra

__Abstract:__

Let F be a field, and fix an element q of F. The q-deformed Heisenberg algebra H(q) is the unital associative algebra over F with generators A, B and relation which asserts that AB - qBA is the unity element. By the set of all Lie polynomials or undeformed commutators in A, B, we mean the Lie subalgebra L of H(q) generated by A, B. We present results from several studies that describe such Lie polynomials. We fully describe L in terms of a basis and a corresponding commutator table, which vary according to cases based on the parameter q. If F is the complex field, and if q is in the interval (0,1), then H(q) is isomorphic to an algebra of operators on some sequence space such that all the compact operators are in L, and the image of H(q) in the Calkin algebra is an algebra of Laurent polynomials in one indeterminate.

**October 5****, 2018, Friday, 15.30-16.30**

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

__Speaker__: Jakob Palmkvist, Chalmers University of Technology, Theoretical Physics, Göteborg

__Title:__ Generators and relations for (generalized) Cartan superalgebras

__Abstract:__

In Kac's classification of finite-dimensional Lie superalgebras, the contragredient ones can be constructed from Dynkin diagrams similar to those of the simple finite-dimensional Lie algebras, but with additional types of nodes. For example, A(0,n-1)=sl(1|n) can be constructed by adding a "gray'' node to the Dynkin diagram of A_{n-1}=sl(n), corresponding to an odd null root. The Cartan superalgebras constitute a different class, where the simplest example is W(n), the derivation algebra of the Grassmann algebra on n generators. I will in my talk present a novel construction of W(n), from the same Dynkin diagram as A(0,n-1), but with additional generators and relations. I will then generalize this result to the exceptional Lie algebras E_n, which can be extended to infinite-dimensional Borcherds superalgebras, in the same way as A_{n-1} can be extended to A(0,n-1). In this case, the construction leads to so called tensor hierarchy algebras, which provide an underlying algebraic structure of certain supergravity models related to string theory.

**October 10, 2018, Wednesday, 15.30-16.30**

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

__Speaker__: Markku Jääskeläinen, Senior Lecturer in Physics, UKK, Mälardalen University

__Title:__ Exploring many-body dynamics of Ultracold gases in confined quantum wells and the quantum physics of self-gravitating microspheres.

**Obs!** (Lecture 2, including modelling and numerical simulations for physical models mentioned in the abstract, will be made independent and accesible for those who missed first September 12 lecture. So, all are wellcome!)

__Abstract:__

In this talk I briefly sketch two of the topics I have worked on since finishing my PhD at the Royal Institute of Technology.

The first line of research was largely motivated by a lack of understanding for experimental results at Stanford University on Bose-Einstein condensates of rubidium atoms in optical lattice traps. Simplified spherical-cow type modelling and numerical simulations turned out to be successful, eventually even convincing the experimentalists.

My research in the second area grew out of my attempts of coming to terms with what reality, if any, quantum mechanics actually describes and how gravity at the classical level fits into the picture. Central here are investigations of how non-relativistic gravity influences quantum phenomena of small solid objects. The subject has been attracting increasing interest in recent years and some basic foundational questions remain open. My efforts go inte tackling these with mathematical rather than philosophical tools in order to offer the experimental community something to observe or refute.

**November 21****, 2018, Wednesday, 15.15-16.05**

(MAM workshop in Probability theory, Mathematical Statistics and Applications)

__Location__: U2-158, Västerås, Mälardalen University

__Speaker__: Professor Nikolai Leonenko, Cardiff University, United Kingdom

__Title:__ Fractional Poisson random fields and martingales

__Abstract:__

We present new properties for the Fractional Poisson process [1,2,3,6,9], Fractional non-homogeneous Poisson process [7,8], Fractional Poisson fields on the plane [1,5]. A martingale characterization for Fractional Poisson processes is given. We extend this result to Fractional Poisson fields, obtaining some other characterizations. The fractional differential equations are studied. The covariance structure is given. Finally, we give some simulations of the Fractional Poisson fields on the plane.

*Joint work with* G. Aletti (University of Milan, Italy) and E. Merzbach (Bar Ilan University, Israel).

*References*

[1] Aletti, G., Leonenko, N.N. and Marzbach, E. (2018) Fractional Poisson fields and martingales, Journal of Statistical Physics, 170, N4, 700-730

[2] Beghin,L. and E. Orsingher, E. (2009) Fractional Poisson processes and related planar random motions, Electron. J. Probab. 14, no. 61, 1790--1827.

[3] Beghin,L. and E. Orsingher, E. (2010) Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron. J. Probab. 15, no. 22, 684--709.

[4] Kerss, A.D.J., Leonenko, N.N. and and Sikorskii, A. (2014) Fractional Skellam processes with applications to finance, Fractional Calculus and Applied Analysis, 17. No.2, pp 532-551

[5] Leonenko, N.N. and Merzbach, E.(2015) Fractional Poisson fields, Methodology and Computing in Applied Probability, 17, 155-168

[6] Leonenko, N.N., Meerschaert, M.M., Schilling, R.L. and Sikorskii, A. (2014) Correlation structure of time-changed Lévy processes, Commun. Appl. Ind. Math. 6, no. 1, e-483, 22 pp.

[7] Leonenko, N., Scalas, E. and Trinh, M. (2017) The fractional non-homogeneous Poisson process, Statistics and Probability Letters, 120, 147-156

[8] Leonenko, N., Scalas, E. and Trinh, M. (2017) Limit theorems for the fractional non-homogeneous Poisson process, submitted, https://arxiv.org/abs/1711.08768

[9] Meerschaert, M.M., Nane, E. and Vellaisamy, P. (2011) The fractional Poisson process and the inverse stable subordinator, Electron. J. Probab. 16, no. 59, 1600-1620.

**November 21****, 2018, Wednesday, 16.10-17.00**

(MAM workshop in Probability theory, Mathematical Statistics and Applications)

__Location__: U2-158, Västerås, Mälardalen University

__Speaker__: Professor Wolfgang Schmid, European University, Frankfurt (Oder), Germany

__Title:__ Spatial and Spatio-Temporal Nonlinear Processes

__Abstract:__

In this talk a new spatial model is presented that incorporates heteroscedastic variances depending on neighboring locations. The proposed process is regarded as the spatial equivalent of the temporal autoregressive conditional heteroscedastic (ARCH) model. It is shown as well how the introduced spatial ARCH model can be used in spatiotemporal settings. The process turns out to be strictly and weakly stationary under some conditions on the noise process and the weight matrix. Although it possesses several properties of a temporal ARCH process some important features are no longer fulfilled. The conditional variance of the process depends on all locations and only for an oriented process it is a function of the locations lying closer to the center. Moreover, the squared process is not a spatial autoregressive process. In order to estimate the parameters of the spatial process the maximum-likelihood approach is applied. For a certain type of weight matrices it is proved that the estimators are asymptotic normally distributed. Via Monte Carlo simulations, the performance of the estimator for various spatial weighting matrices and for a finite sample size is analyzed. Moreover, the well known spatial autoregressive model is combined with the spatial ARCH model assuming heteroscedastic errors. Eventually, the proposed autoregressive process is illustrated using an empirical example. Specifically, the lung cancer mortality in 3108 U.S. counties is modeled and the introduced model is compared with two benchmark approaches.

**December 12 , 2018, Wednesday, 15.30-16.30**

__Location__: U2-158, Västerås, Mälardalen University

__Speaker__: Antonio Possolo (National Institute of Standards and Technology, USA)

https://www.nist.gov/people/antonio-possolo

__co-author__: Olha Bodnar (Division of Applied Mathematics, MAM, Malardalen University, Sweden)

__Title:__ Approximate Bayesian Evaluations of Measurement Uncertainty

__Abstract:__

The *Guide to the expression of uncertainty in measurement *(GUM) includes formulas that produce an estimate of a scalar output quantity that is a function of several input quantities, and an approximate evaluation of the associated standard uncertainty.

This contribution presents approximate, Bayesian counterparts of those formulas for the case where the output quantity is a parameter of the joint probability distribution of the input quantities, also taking into account any information about the value of the output quantity available prior to measurement expressed in the form of a probability distribution on the set of possible values for the measurand.

The approximate Bayesian estimates and uncertainty evaluations that we present have a long history and illustrious pedigree, and provide sufficiently accurate approximations in many applications, yet are very easy to implement in practice. Differently from exact Bayesian estimates, which involve either (analytical or numerical) integrations, or Markov Chain Monte Carlo sampling, the approximations that we describe involve only numerical optimization and simple algebra. Therefore, they make Bayesian methods widely accessible to metrologists.

The proposed techniques will be illustrated in several instances of application, selected from among the following: isotopic ratio of silver in a commercial silvernitrate; odds of cryptosporidiosis in AIDS patients; height of a manometer column; mass fraction of chromium in a reference material; and potential-difference in a Zener voltage standard.

**REFERENCE**

A. Possolo and O. Bodnar (2018) *Metrologia *55(2), 147–157 DOI 10.1088/1681-7575/aaa5be