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• 1.
University Pretoria, South Africa.
University of the Basque Country, Spain. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Common Fixed Points of Generalized Rational Type Cocyclic Mappings in Multiplicative Metric Spaces2015In: Discrete dynamics in nature and society, ISSN 1026-0226, E-ISSN 1607-887X, Vol. 2015, article id 532725Article in journal (Refereed)

The aim of this paper is to present fixed point result of mappings satisfying a generalized rational contractive condition in the setup of multiplicative metric spaces. As an application, we obtain a common fixed point of a pair of weakly compatible mappings. Some common fixed point results of pair of rational contractive types mappings involved in cocyclic representation of a nonempty subset of a multiplicative metric space are also obtained. Some examples are presented to support the results proved herein. Our results generalize and extend various results in the existing literature.

• 2.
Ton Duc Thang University, Ho Chi Minh City, Viet Nam.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. University of Kragujevac, Serbia. University of Belgrade, Belgrade.
On Weakly Commuting Set-Valued Mappings on a Domain of Sets Endowed with Directed Graph2017In: Results in Mathematics, ISSN 1422-6383, Vol. 71, no 3-4, p. 1277-1295Article in journal (Refereed)

The aim of this paper is to present coincidence and common fixed point results of set-valued mappings satisfying certain generalized graphic F-contractive conditions on a family of sets endowed with a graph. It is worth mentioning that these results are obtained without appealing to any form of continuity of mappings involved herein. Some examples are presented to support the results proved in this paper. Our results unify, generalize and extend various comparable results in the existing literature. Â© 2016, Springer International Publishing.

• 3.
Ton Duc Thang Univ, Ho Chi Minh City, Vietnam..
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Fac Sci, Dept Math, Radoja Domanovica 12, Kragujevac 34000, Serbia. State Univ Novi Pazar, Serbia..
Common fixed points of set-valued F-contraction mappings on domain of sets endowed with directed graph2017In: Computational and Applied Mathemathics, ISSN 0101-8205, Vol. 36, no 4, p. 1607-1622Article in journal (Refereed)

The aim of this paper is to present common fixed point results of set-valued graphic F-contraction mappings on a family of sets endowed with a graph. Some examples are presented to support the results proved herein. Our results unify, generalize and extend various results in the existing literature.

• 4.
École polytechnique fédérale de Lausanne, Switzerland.
École polytechnique fédérale de Lausanne, Switzerland. École polytechnique fédérale de Lausanne, Switzerland.
Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems2019In: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 357, no 6, p. 545-551Article in journal (Refereed)

This paper presents two new approaches for finding the homogenized coefficients of multiscale elliptic PDEs. Standard approaches for computing the homogenized coefficients suffer from the so-called resonance error, originating from a mismatch between the true and the computational boundary conditions. Our new methods, based on solutions of parabolic and elliptic cell problems, result in an exponential decay of the resonance error.

• 5.
Mälardalen University, School of Education, Culture and Communication.
Mälardalen University, School of Education, Culture and Communication.
Black-Litterman Model: Practical Asset Allocation Model Beyond Traditional Mean-Variance2016Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

This paper consolidates and compares the applicability and practicality of Black-Litterman model versus traditional Markowitz Mean-Variance model. Although well-known model such as Mean-Variance is academically sound and popular, it is rarely used among asset managers due to its deficiencies. To put the discussion into context we shed light on the improvement made by Fisher Black and Robert Litterman by putting the performance and practicality of both Black- Litterman and Markowitz Mean-Variance models into test. We will illustrate detailed mathematical derivations of how the models are constructed and bring clarity and profound understanding of the intuition behind the models. We generate two different portfolios, composing data from 10-Swedish equities over the course of 10-year period and respectively select 30-days Swedish Treasury Bill as a risk-free rate. The resulting portfolios orientate our discussion towards the better comparison of the performance and applicability of these two models and we will theoretically and geometrically illustrate the differences. Finally, based on extracted results of the performance of both models we demonstrate the superiority and practicality of Black-Litterman model, which in our particular case outperform traditional Mean- Variance model.

• 6.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, School of Physical Sciences, Makerere University, Kampala, Uganda.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam,Tanzania. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, Makerere University, Kampala, Uganda. Department of Mathematics, Makerere University, Kampala, Uganda. Department of Mathematics, Makerere University, Kampala, Uganda. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
A Variant of Updating Page Rank in Evolving Tree graphs2019In: Proceedings of 18th Applied Stochastic Models and Data Analysis International Conference with the Demographics 2019 Workshop, Florence, Italy: 11-14 June, 2019 / [ed] Christos H. Skiadas, ISAST: International Society for the Advancement of Science and Technology , 2019, p. 31-49Conference paper (Refereed)

PageRank update refers to the process of computing new PageRank values after change(s) (addition or removal of links/vertices) has occurred in real life networks. The purpose of the updating is to avoid recalculating the values from scratch. To efficiently carry out the update, we consider PageRank as the expected number of visits to target vertex if multiple random walks are performed, starting at each vertex once and weighing each of these walks by a weight value. Hence, it might be looked at as updating non-normalised PageRank. In the proposed approach, a scaled adjacency matrix is sequentially updated after every change and the levels of the vertices being updated as well. This enables sets of internal and sink vertices dependent on their roots or parents, thus vector-vector product can be performed sequentially since there are no infinite steps from one vertex to the other.

• 7.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, School of Physical Sciences, Makerere University, Kampala, Uganda.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam,Tanzania. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, School of Physical Sciences, Makerere University, Kampala, Uganda. Department of Mathematics, School of Physical Sciences, Makerere University, Kampala, Uganda. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
PageRank in evolving tree graphs2018In: Stochastic Processes and Applications: SPAS2017, Västerås and Stockholm, Sweden, October 4-6, 2017 / [ed] Sergei Silvestrov, Anatoliy Malyarenko, Milica Rančić, Springer, 2018, Vol. 271, p. 375-390Chapter in book (Refereed)

In this article, we study how PageRank can be updated in an evolving tree graph. We are interested in finding how ranks of the graph can be updated simultaneously and effectively using previous ranks without resorting to iterative methods such as the Jacobi or Power method. We demonstrate and discuss how PageRank can be updated when a leaf is added to a tree, at least one leaf is added to a vertex with at least one outgoing edge, an edge added to vertices at the same level and forward edge is added in a tree graph. The results of this paper provide new insights and applications of standard partitioning of vertices of the graph into levels using breadth-first search algorithm. Then, one determines PageRanks as the expected numbers of random walk starting from any vertex in the graph. We noted that time complexity of the proposed method is linear, which is quite good. Also, it is important to point out that the types of vertex play essential role in updating of PageRank.

• 8.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, School of Physical Sciences, Makerere University, Kampala, Uganda.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam,Tanzania. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Evaluation of Stopping Criteria for Ranks in Solving Linear Systems2019In: Data Analysis and Applications 1: Clustering and Regression, Modeling‐estimating, Forecasting and Data Mining, Volume 2 / [ed] Christos H. Skiadas, James R. Bozeman, John Wiley & Sons, 2019, Chapter 10, p. 137-152Chapter in book (Refereed)

Bioinformatics, internet search engines (web pages) and social networks are some of the examples with large and high sparsity matrices. For some of these systems, only the actual ranks of the solution vector is interesting rather than the vector itself. In this case, it is desirable that the stopping criterion reflects the error in ranks rather than the residual vector that might have a lower convergence. This chapter evaluates stopping criteria on Jacobi, successive over relaxation (SOR) and power series iterative schemes. Numerical experiments were performed and results show that Kendall's correlation coefficient gives good stopping criterion of ranks for linear system of equations. The chapter focuses on the termination criterion as means of obtaining good ranks. It outlines some studies carried out on stopping criteria in solving the linear system.

• 9.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, School of Physical Sciences, Makerere University, Kampala, Uganda.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam,Tanzania. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Stockholm University, Sweden. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, Makerere University, Kampala, Uganda. Department of Mathematics, Makerere University, Kampala, Uganda.
Nonlinearly  Perturbed Markov Chains  and  Information Networks2019In: Proceedings of 18th Applied Stochastic Models and Data Analysis International Conference with the Demographics 2019 Workshop, Florence, Italy: 11-14 June, 2019 / [ed] Christos H. Skiadas, ISAST: International Society for the Advancement of Science and Technology , 2019, p. 51-79Conference paper (Refereed)

The paper is devoted to studies of perturbed Markov chains commonly used for description of information networks. In such models, the matrix of transition probabilities for the corresponding Markov chain is usually regularised by adding  a special damping matrix multiplied by a small damping (perturbation) parameter ε. In this paper, we present results of the detailed perturbation analysis of Markov chains with damping component and numerical experiments supporting and illustrating the results of this perturbation analysis.

• 10. Abramov, V.
Paal, E.Tallinn University of Technology.Silvestrov, Sergei D.Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.Stolin, A.Chalmers University of Techology.
Proceedings of the 3rd Baltic-Nordic Workshop “Algebra, Geometry, and Mathematical Physics”2008Conference proceedings (editor) (Refereed)
• 11. Abramov, Viktor
Mälardalen University, School of Education, Culture and Communication.
Preface [Special issue devoted to the 4th Baltic-Nordic Workshop “Algebra, Geometry and Mathematical Physics”]2010In: Proceedings of the Estonian Academy of Sciences, ISSN 1736-6046, E-ISSN 1736-7530, Vol. 59, no 4Article in journal (Refereed)
• 12.
Mälardalen University, School of Education, Culture and Communication.
Analyzing the "True" Delta, Gamma and Vega of European Swaptions in Black-76 and Bachelier Models Using Python2019Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

Option pricing models such as the Black-76, Bachelier and Stochastic Alpha Beta Rho (SABR) models define delta, gamma and vega using the forward rate as the underlying instrument. In terms of options on swaps, this underlying instrument is the forward swap rate, which is the market fixed rate on the swap. This work employs the Black-76 and Bachelier models to determine delta, gamma and vega of a European swaption, but rather uses the swap value as the underlying instrument. We use data in our implementation and the work is done in the python programming language.

Our results yielded relatively higher absolute delta values than those implied by the conventional Black-76 and Bachelier models. This means that our method yields relatively higher sensitivity of the swaption value to small changes in underlying asset value. It also means higher trading volumes of the swap contract to hedge against small changes in the value of the underlying swap using our method. It was also observed from our gamma values that both our method and the conventional Black-76 and Bachelier models can provide better sensitivities, relative to each other, to big changes in the underlying swap value. This, however, depends on the choice of strike rate. Further, our work produced comparatively lower absolute vega values, hence, lower sensitivity to changing implied volatility. To be able to use volatility values in both the normal and log-normal sense, we converted from normal to log-normal volatilities. This was achieved numerically using the Newton Raphson method implemented in python.

Changes in swap value and volatility were mimicked using basis point additions and subtractions of certain parameter values, specifically, floating rates, forward rate and volatility values. Each basis point adjustment in the necessary parameter yielded a different delta, gamma or vega value. Since the swap value or volatility can change multiple times within a specific time period, it was observed that there can exist a series of delta, gamma and vega values within a specific period of time.

• 13.
Mälardalen University, Department of Mathematics and Physics.
Java Applet for the Pricing of Exotic Options by Monte-Carlo Simulations in a Levy market with Stochastic Volatility2006Independent thesis Basic level (professional degree), 10 points / 15 hpStudent thesis
• 14.
Mälardalen University, School of Education, Culture and Communication.
MARGIN VALUATION ADJUSTMENT of an INTEREST RATE SWAP2019Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

After the last financial crisis, regulators started to strengthen the measures to reduce the risk on the financial market by requiring margin to be posted. In addition to the Variation Margin that reflects the daily change in market value of the contracts, Initial Margin has to be posted to compensate the fluctuation of the collateral value. The Margin Valuation Adjustment (MVA) represents the funding price of the initial margin that has to be calculated. The numerical calculation of the MVA is a real challenge as it requires to run a Monte Carlo simulation inside another Monte Carlo simulation. However, alternative formula could be used to approximate the result of the MVA. This thesis highlights the essential notion about the MVA and the available tools to calculate its value.

• 15.
Mälardalen University, School of Education, Culture and Communication.
Risk-Efficient Portfolios; Estimation Error In Essence2016Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

This thesis primarily looks at estimation error problems and other related issues arising in connection with portfolio optimization. With some available assets, a portfolio program or optimizer seeks to distribute a fixed amount of capital among these available assets to optimize some cost function. In this regard, Markowitz portfolio selection basis defines the variance of the portfolio return to being that of the portfolio risk and tries to find an allocation that reduces or minimizes the risk subject to a target mean or expected return. Should the mean return vector and the covariance matrix of returns for the underlying assets be known, the Markowitz problem is said to have a closed-form solution.

In practice, however, an estimation is made from historical data for unknown expected returns and the covariance matrix of the returns, and this brings into the domain several problems such as estimation problems and renders the Markowitz theory impracticable in real-life portfolio applications. Estimators necessary to remedy these problems would be made bare to show how possible it is to tackle such issues.

In the concept demonstration sections, the analysis starts with the price data of 40 stocks and the S\&P index. The efficient frontier is introduced and used to show how the estimators take effect.

Finally, implementation is made possible using the R Programming Language to demonstrate the necessary concepts with the conclusion presented at the end.

• 16.
Mälardalen University, Department of Mathematics and Physics.
Lärares syn på lässvaga elever inom matematik2008Independent thesis Basic level (degree of Bachelor), 10 points / 15 hpStudent thesis

Syftet med detta arbete är att ta reda på hur lärarna upplever att lässvaga elever påverkas av sina lässvårigheter inom matematiken samt hur lärarna hanterar de problem som uppstår för att hjälpa eleven. Jag har valt den kvalitativa arbetsmetoden med semistrukturerade intervjufrågor. Resultatet visar att svag läsförmåga påverkar eleven inom matematiken men också inom andra ämnen. Inom matematiken påverkas eleven särskilt vid lästal samt vid problemlösning om eleven lämnas ensam med en text som den inte klarar av att läsa eller tyda. För att stödja eleven kan läraren eller någon annan läsa texten eller så arbetar eleverna i grupp där samarbetet mot resultatet är viktigast, inte att kunna läsa texten. Slutsatsen visar att eleven aldrig bör lämnas ensam med en text som den inte förstår, då eleven utestängs från den matematiska processen och det matematiska tänkandet.

• 17.
Mälardalen University, Department of Mathematics and Physics.
Problemlösning, kontext och kompetens2007Independent thesis Basic level (degree of Bachelor), 10 points / 15 hpStudent thesis

Syftet med arbetet är att undersöka hur matematiklärarstudenter ska introducera problemlösning för sina elever och vad som påverkar studenternas tolkningar av problem. Jag har även undersökt vilka kompetenser studenterna anser att en matematiklärare behöver ha för att genomföra en framgångsrik undervisning i matematik genom problemlösning. I undersökningen diskuterar tre grupper med matematiklärarstudenter hur de ska introducera ett problem för eleverna och vilka komprtenser de anser att en matematiklärare behöver ha för att undervisa i matematik genom problemlösning. Resultaten visar att studenterna har olika strategier för att introducera problemlösning för eleverna och att studenternas tolkningar av problemet påverkar introduktionen. Studenterna gör olika tolkningar av problemet beroende av i vilken kontext de tolkar problemet. De har ingen enhetlig bild av vilka kompetenser som krävs av en matematiklärare, för att genomföra en undervisning som svarar mot skolverkets mål.

• 18.
Mälardalen University, School of Education, Culture and Communication.
Mälardalen University, School of Education, Culture and Communication.
Implied Volatility Surface Approximation under a Two-Factor Stochastic Volatility Model2018Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

Due to recent research disproving old claims in ﬁnancial mathematics such as constant volatility in option prices, new approaches have been incurred to analyze the implied volatility, namely stochastic volatility models. The use of stochastic volatility in option pricing is a relatively new and unexplored ﬁeld of research with a lot of unknowns, where new answers are of great interest to anyone practicing valuation of derivative instruments such as options. With both single and two-factor stochastic volatility models containing various correlation structures with respect to the asset price and differing mean-reversions of variance the question arises as to how these values change their more observable counterpart: the implied volatility. Using the semi-analytical formula derived by Chiarella and Ziveyi, we compute European call option prices. Then, through the Black–Scholes formula, we solve for the implied volatility by applying the bisection method. The implied volatilities obtained are then approximated using various models of regression where the models’ coefﬁcients are determined through the Moore–Penrose pseudo-inverse to produce implied volatility surfaces for each selected pair of correlations and mean-reversion rates. Through these methods we discover that for different mean-reversions and correlations the overall implied volatility varies signiﬁcantly and the relationship between the strike price, time to maturity, implied volatility are transformed.

• 19.
Mälardalen University, School of Education, Culture and Communication. Kvinna.
Mälardalen University, School of Education, Culture and Communication.
Elever med matematikängslan: En kvalitativ studie om matematiklärares syn på matematikängslan samt hur de arbetar för att förebygga detta hos elever2019Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis

I denna studie undersöktes det hur matematiklärare i mellanstadiet uppfattar fenomenet matematikängslan samt vilka faktorer de anser påverkar elevers inställning kring matematik. Vidare syftade studien till att inhämta kunskaper om hur dessa lärare arbetar för att förebygga att matematikängslan uppstår bland elever. Precis som när elever kan känna sig ängsliga över andra saker, upplever elever med matematikängslan att de är spända och att de inte är kapabla till att genomföra olika matematiska moment (Dowker, 2016). Studien är baserad på den socialkognitiva teorin som är grundad av Albert Bandura (2012). Han menar att elevernas affektiva upplevelser och lärande uppstår när människor observerar och imiterar andra, samtidigt som de är uppmärksamma, använder sitt minne och har motivation (Bandura, 2012). För att svara på våra forskningsfrågor har vi haft semistrukturerade intervjuer med möjlighet för följdfrågor. Informanterna beskrev begreppet matematikängslan som en känsla där eleverna känner sig oroliga och frånvarande från lektionerna. Resultatet visar att de faktorer som påverkar elevers matematikängslan är individanpassning, inställning, erfarenhet, kön och ålder. Vidare beskrev informanterna att matematikängslan kan förebyggas genom att läraren bland annat skapar ett tryggt och lärorikt lärandeklimat samt har en positiv inställning till matematikämnet.

• 20.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University. Mälardalen University.
Asymptotics of Implied Volatility in the Gatheral Double Stochastic Volatility Model2019In: Proceedings of 18th Applied Stochastic Models and Data Analysis International Conference with the Demographics 2019 Workshop, Florence, Italy: 11-14 June, 2019 / [ed] Christos H. Skiadas, ISAST: International Society for the Advancement of Science and Technology , 2019, p. 81-90Conference paper (Refereed)

The double-mean-reverting model by Gatheral [1] is motivated by empirical dynamics of the variance of the stock price. No closed-form solution for European option exists in the above model. We study the behaviour of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the- money. Using the method by Pagliarani and Pascucci [6], we calculate explicitly the first few terms of the asymptotic expansion of the implied volatility within a parabolic region.

• 21.
Mälardalen University, Department of Mathematics and Physics.
Laborativt material i matematikundervisningen - med fokus på geobrädet2006Independent thesis Basic level (degree of Bachelor), 10 points / 15 hpStudent thesis

yftet med detta arbete var att undersöka hur geobrädet fungerar i praktiken. För att få en inblick i hur det kan vara att arbeta med geobrädet, valde jag att gå ut på min partnerskola i 2 och utföra en kvalitativ undersökning. Denna undersökning bestod i att hålla ett antal lektioner där geobrädet var i fokus. Jag har också utfört en litteraturstudie för att få bättre inblick i det laborativa materialets betydelse för elevers matematikutveckling. Resultatet av min undersökning visade att eleverna blev mycket engagerade och aktiva och att de tyckte att arbetet med geobrädet var roligt och spännande. Jag som blivande lärare lärde mig också mycket om hur jag kan använda geobrädet i mitt kommande yrke och vad som är bra att tänka på när man arbetar med detta laborativa material. De slutsatser som jag kan dra efter att ha utfört detta examensarbete är att elever verkligen uppskattar att få laborera och prova saker på egen hand. Dessutom har jag som blivande pedagog blivit ännu mer övertygad om att det laborativa materialet har stor betydelse för elevernas matematiska utveckling.

• 22.
Mälardalen University, School of Education, Culture and Communication.
Mälardalen University, School of Education, Culture and Communication.
Matematik i förskolan: - ur förskollärares perspektiv2013Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis

Syftet med denna studie var att utforska förskollärares perspektiv på arbetet med matematiken i förskolan. Studien har utforskat hur förskollärare arbetar med matematik för att tillvarata, synliggöra och främja barns lärande och utveckling i förskolans verksamhet. Detta är en kvalitativ studie där sex stycken förskollärare intervjuats för att ta reda på deras perspektiv kring arbetet med att synliggöra och tillvarata matematik för att främja barnens lärande och utveckling. Resultatet visade att matematiken har en naturlig del i förskollärares vardagsarbete och att matematiska begrepp var en central del i barns lärande och utveckling i matematik. Att skapa möjligheter att uppleva matematik på ett positivt sätt har förskollärarna uttryckt som betydelsefullt. Förskollärarna i studien har beskrivit att barnen redan i förskolan kan ges förutsättning till ett lustfyllt lärande i matematik som grund för vidare lärande. Slutsatser som kan ställas utifrån denna studie är att en medvetenhet kring arbetet med matematik i förskolan är viktigt för att främja barns lärande och utveckling i matematik.

• 23.
Izmir University, Izmir, Turkey. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
On the global solution of a fuzzy linear system2014In: Journal of Fuzzy Set Valued Analysis, ISSN 2193-4169, article id jfsva-00190Article in journal (Other academic)

The global solution of a fuzzy linear system contains the crisp vector solution of a real linear system. So discussion about the global solution of a $n\times n$ fuzzy linear system $A\tilde{x}=\tilde{b}$ with a fuzzy number vector $b$ in the right hand side and crisp a coefficient matrix $A$ is considered. The advantage of the paper is developing a new algorithm to find the solution of such system by considering a global solution based upon the concept of a convex fuzzy numbers. At first the existence and uniqueness of the solution are introduced and then the related theorems and properties about the solution are proved in details. Finally the method is illustrated by solving some numerical examples.

• 24.
Mälardalen University, School of Education, Culture and Communication.
Nonlinear Least-Square Curve Fitting of Power-Exponential Functions: Description and comparison of different fitting methods2017Independent thesis Advanced level (degree of Master (Two Years)), 80 credits / 120 HE creditsStudent thesis

This thesis examines how to find the best fit to a series of data points when curve fitting using power-exponential models. We describe the different numerical methods such as the Gauss-Newton and Levenberg-Marquardt methods to compare them for solving non-linear least squares of curve fitting using different power-exponential functions. In addition, we show the results of numerical experiments that illustrate the effectiveness of this approach.Furthermore, we show its application to the practical problems by using different sets of data such as death rates and rocket-triggered lightning return strokes based on the transmission line model.

• 25.
University of California, Berkeley, USA.
Mälardalen University, Department of Mathematics and Physics. University of Latvia, Latvia. University of Latvia, Latvia. University of Bonn, Germany.
Quantum finite multitape automata1999In: SOFSEM’99: Theory and Practice of Informatics: 26th Conference on Current Trends in Theory and Practice of Informatics Milovy, Czech Republic, November 27 — December 4, 1999 Proceedings, 1999, Vol. 1725, p. 340-348Conference paper (Refereed)

Quantum finite automata were introduced by C. Moore, J. P. Crutchfield [4], and by A. Kondacs and J. Watrous [3]. This notion is not a generalization of the deterministic finite automata. Moreover, in [3] it was proved that not all regular languages can be recognized by quantum finite automata. A. Ambainis and R. Freivalds [1] proved Chat for some languages quantum finite automats may be exponentially more concise rather than both deterministic and probabilistic finite automata. In this paper we introduce the notion of quantum finite multi-tape automata and prove that there is a language recognized by a quantum finite automaton but not by deterministic or probabilistic finite automats. This is the first result on a problem which can be solved by a quantum computer but not by a deterministic or probabilistic computer. Additionally we discover unexpected probabilistic automata recognizing complicated languages.

• 26.
Faculté des Sciences, Université de Sfax, Tunisia .
Université de Haute Alsace, France . Lund University.
Ternary q-Virasoro-Witt Hom-Nambu-Lie algebras2010In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 43, no 26, p. 265204-Article in journal (Refereed)

In this paper we construct ternary q-Virasoro-Witt algebras which q-deform the ternary Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos using su(1, 1) enveloping algebra techniques. The ternary Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos depend on a parameter and are not Nambu-Lie algebras for all but finitely many values of this parameter. For the parameter values for which the ternary Virasoro-Witt algebras are Nambu-Lie, the corresponding ternary q-Virasoro-Witt algebras constructed in this paper are also Hom-Nambu-Lie because they are obtained from the ternary Nambu-Lie algebras using the composition method. For other parameter values this composition method does not yield a Hom-Nambu-Lie algebra structure for q-Virasoro-Witt algebras. We show however, using a different construction, that the ternary Virasoro-Witt algebras of Curtright, Fairlie and Zachos, as well as the general ternary q-Virasoro-Witt algebras we construct, carry a structure of the ternary Hom-Nambu-Lie algebra for all values of the involved parameters.

• 27.
Mälardalen University, School of Education, Culture and Communication.
Sustainability for Portfolio Optimization2019Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

The 2007-2008 financial crash and the looming climate change and global warming have heightened interest in sustainable investment. But whether the shift is as a result of the financial crash or a desire to preserve the environment, a sustainable investment might be desirable. However, to maintain this interest and to motivate investors in indulging in sustainability, there is the need to show the possibility of yielding positive returns. The main objective of the thesis is to investigate whether the sustainable investment can lead to higher returns. The thesis focuses primarily on incorporating sustainability into Markowitz portfolio optimization. It looks into the essence of sustainability and its impact on companies by comparing different concepts. The analysis is based on the 30 constituent stocks from the Dow Jones industrial average or simply the Dow. The constituents stocks of the Dow, from 2007-12-31 to 2018-12-31 are investigated. The thesis compares the cumulative return of the Dow with the sustainable stocks in the Dow based on their environmental, social and governance (ESG) rating. The results are then compared with the Dow Jones Industrial Average denoted by the symbol (^DJI) which is considered as the benchmark for my analysis. The constituent stocks are then optimized based on the Markowitz mean-variance framework and a conclusion is drawn from the constituent stocks, ESG, environmental, governance and social asset results. It was realized that the portfolio returns for stocks selected based on their environmental and governance ratings were the highest performers. This could be due to the fact that most investors base their investment selection on the environmental and governance performance of companies and the demand for stocks in that category could have gone up over the period, contributing significantly to their performance.

• 28.
Mälardalen University, Department of Mathematics and Physics.
En analys av en mattestuga: - elever med matematiksvårigheter2007Independent thesis Basic level (degree of Bachelor), 10 points / 15 hpStudent thesis
• 29.
Mälardalen University, School of Education, Culture and Communication.
Valuation Impact on a Plain Vanilla Swap due to Transition from IBOR2019Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 30.
WorldLight.com AB, Sweden.
Mälardalen University, School of Education, Culture and Communication.
The mathematics of internet search engines2008In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, Vol. 104, no 2, p. 211-242Article in journal (Refereed)

This article presents a survey of techniques for ranking results in search engines, with emphasis on link-based ranking methods and the PageRank algorithm. The problem of selecting, in relation to a user search query, the most relevant documents from an unstructured source such as the WWW is discussed in detail. The need for extending classical information retrieval techniques such as boolean searching and vector space models with link-based ranking methods is demonstrated. The PageRank algorithm is introduced, and its numerical and spectral properties are discussed. The article concludes with an alternative means of computing PageRank, along with some example applications of this new method.

• 31.
Mälardalen University, School of Education, Culture and Communication.
Mälardalen University, School of Education, Culture and Communication.
Lyfts frågorna i lyftet?: En studie i hur Skolverket modellerar frågor i Matematiklyftet2019Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis

Tidigare studier av matematik visar att lärare ofta ställer frågor som eftersöker korta svar.Detta gör att det saknas frågor som fokuserar på att eleverna resonerar, analyserar ochkommunicerar. Dessa typer av frågor bidrar till att nå en större matematisk utveckling föreleverna. Den här studien observerade lektionsfilmer från fortbildningen Matematiklyftetårskurs 4-6 med syfte i att undersöka vilka typer av frågor som ställdes. Resultatet avstudien visade att det även här fanns en stor andel frågor fokuserade på korta svar där justsvaret är det viktiga. Dock ställdes det fler frågor som byggde på att skapa diskussion ochför att eleverna skulle få utveckla sina tankegångar. Studiens slutsats är attMatematiklyftet vill arbeta mot att bryta den rådande klassrumssituationen och iställetskapa ett mer kommunicerande och resonerande klassrum.

• 32.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
University of Copenhagen.
On the universality of the Epstein zeta functionManuscript (preprint) (Other academic)

We study universality properties of the Epstein zeta function En(L,s) for lattices L of large dimension n and suitable regions of complex numbers s . Our main result is that, as n→∞ , En(L,s) is universal in the right half of the critical strip as L varies over all n -dimensional lattices L . The proof uses an approximation result for Dirichlet polynomials together with a recent result on the distribution of lengths of lattice vectors in a random lattice of large dimension and a strong uniform estimate for the error term in the generalized circle problem. Using the same basic approach we also prove that, as n→∞ , En(L1,s)−En(L2,s) is universal in the full half-plane to the right of the critical line as (L1,L2) varies over all pairs of n -dimensional lattices. Finally, we prove a more classical universality result for En(L,s) in the s -variable valid for almost all lattices L of dimension n . As part of the proof we obtain a strong bound of En(L,s) on the critical line that is subconvex for n≥5 and almost all n -dimensional lattices L.

• 33.
Mälardalen University, School of Education, Culture and Communication.
Mälardalen University, School of Education, Culture and Communication.
Forecasting the Stock Market: A Neural Network Approch2009Student thesis

Forecasting the stock market is a complex task, partly because of the random walk behavior of the stock price series. The task is further complicated by the noise, outliers and missing values that are common in financial time series. Despite of this, the subject receives a fair amount of attention, which probably can be attributed to the potential rewards that follows from being able to forecast the stock market.

Since artificial neural networks are capable of exploiting non-linear relations in the data, they are suitable to use when forecasting the stock market. In addition to this, they are able to outperform the classic autoregressive linear models.

The objective of this thesis is to investigate if the stock market can be forecasted, using the so called error correction neural network. This is accomplished through the development of a method aimed at finding the optimum forecast model.

The results of this thesis indicates that the developed method can be applied successfully when forecasting the stock market. Of the five stocks that were forecasted in this thesis using forecast models based on the developed method, all generated positive returns. This suggests that the stock market can be forecasted using neural networks.

• 34.
Mälardalen University, School of Education, Culture and Communication.
Mälardalen University, School of Education, Culture and Communication.
Hur tänkte du nu?: En studie kring frågetyper i matematikundervisningen2018Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis

• 35.
Riga Technical University, Latvia.
Riga Technical University, Latvia. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Algorithms of the Copula Fit to the Nonlinear Processes in the Utility Industry2017In: Procedia Computer Science, ISSN 1877-0509, E-ISSN 1877-0509, Vol. 104, p. 572-577Article in journal (Refereed)

Our research studies the construction and estimation of copula-based semi parametric Markov model for the processes, which involved in water flows in the hydro plants. As a rule analyzing the dependence structure of stationary time series regressive models defined by invariant marginal distributions and copula functions that capture the temporal dependence of the processes is considered. This permits to separate out the temporal dependence (such as tail dependence) from the marginal behavior (such as fat tails) of a time series. Dealing with utility company data we have found the best copula describing data - Gumbel copula. As a result constructed algorithm was used for an imitation of low probability events (in a hydro power industry) and predictions.

• 36.
Umeå University.
Umeå University. Umeå University.
Fast multiplication of matrices over a finitely generated semiring2008In: Information Processing Letters, ISSN 0020-0190, E-ISSN 1872-6119, Vol. 107, no 6, p. 230-234Article in journal (Refereed)
• 37.
Mälardalen University, Department of Mathematics and Physics.
Mälardalen University, Department of Mathematics and Physics.
Integer Partitions2004Book (Other (popular science, discussion, etc.))
• 38.
Mälardalen University.
Linköping University, Sweden; Mittag Leffler Inst, Djursholm, Sweden. Umeå University, Sweden; Mittag Leffler Inst, Djursholm, Sweden.
Restricted completion of sparse partial Latin squares2019In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 28, no 5, p. 675-695Article in journal (Refereed)

An n × n partial Latin square P is called α-dense if each row and column has at most αn non-empty cells and each symbol occurs at most αn times in P. An n × n array A where each cell contains a subset of {1,.., n} is a (βn, βn, βn)-array if each symbol occurs at most βn times in each row and column and each cell contains a set of size at most βn. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants α, β > 0 such that, for every positive integer n, if P is an α-dense n × n partial Latin square, A is an n × n (βn, βn, βn)-array, and no cell of P contains a symbol that appears in the corresponding cell of A, then there is a completion of P that avoids A; that is, there is a Latin square L that agrees with P on every non-empty cell of P, and, for each i, j satisfying 1 ≤ i, j ≤ n, the symbol in position (i, j) in L does not appear in the corresponding cell of A.

• 39.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, School of Physical Sciences, Makerere University, Kampala, Uganda.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
A comparison of graph centrality measures based on random walks and their computation2019In: Proceedings of 18th Applied Stochastic Models and Data Analysis International Conference with the Demographics 2019 Workshop, Florence, Italy: 11-14 June, 2019 / [ed] Christos H. Skiadas, ISAST: International Society for the Advancement of Science and Technology , 2019, p. 121-135Conference paper (Refereed)

When working with a network it is often of interest to locate the "most important" nodes in the network. A common way to do this is using some graph centrality measures. Since what constitutes an important node is different between different networks or even applications on the same network there is a large amount of different centrality measures proposed in the literature. Due to the large amount of different centrality measures proposed in different fields, there is also a large amount very similar or equivalent centrality measures in the sense that they give the same ranks. In this paper we will focus on centrality measures based on powers of the adjacency matrix or similar matrices and those based on random walk in order to show how some of these are related and can be calculated efficiently using the same or slightly altered algorithms.

• 40.
Mälardalen University, Department of Mathematics and Physics.
Mälardalen University, Department of Mathematics and Physics.
Nationella prov i matematik utan miniräknare?: Med fokus på huvudräkning och miniräknare i anknytning till matematikundervisning och B1-delen i ämnesprovet för år 92008Student thesis

Vi började med att ställa oss frågan varför man valt att utesluta miniräknaren i det obligatoriska Nationella ämnesprovet i matematik för år 9. Enligt PRIM-gruppen, de som konstruerat Nationella prov i matematik, var avsikten att följa målen i styrdokumenten vad gäller huvudräkning och även synpunkter från arbetslivet har bidragit till detta beslut. Vi undrade då vidare hur lärare definierar begreppet huvudräkning och om det finns en generell uppfattning kring miniräknaren som orsak till försämrad huvudräkning. I styrdokumenten står det uttryckligen att eleven skall bruka miniräknaren och använda den som hjälpmedel. Vi har valt en kvalitativ metod i vår studie med intervjuer av lärare och PRIM-gruppen. Alla lärare vi intervjuade hänvisade till styrdokumenten vad gäller den miniräknarfria delen. Vi fann att lärare uppfattar att elever använder miniräknaren ofta även vid enklare beräkningar. Några av de intervjuade lärarna hade speciella lektioner för att träna huvudräkning utan hjälpmedel medans andra ansåg det falla sig naturligt i undervisningen.

• 41.
Mälardalen University, School of Education, Culture and Communication.
Recursive function identification and optimization for ships2019Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

Due to the fact that the fuel consumption of a ship is large, shipowners have a great interest in different kinds of fuel-saving solutions. For any fuel efficiency device, a mathematical function is required, that with high precision is able to give the fuel consumption.

The goal of this thesis is to identify the parameters of this function using the recursive least squares (RLS). These parameters will then be used for making predictions of the fuel consumption for a voyage. The implementation is made in Matlab. We will also investigate the quality of predicting fuel consumption using a neural network, which is implemented in Python.

The results show that RLS is an effective method for making predictions, whereas the neural network could use some further development before any conclusions can be made.

• 42.
École Polytechniques Fédérale de Lausanne, Switzerland.
Uppsala University, Uppsala, Sweden. Uppsala University, Uppsala, Sweden.
Temporal upscaling in micromagnetism via heterogeneous multiscale methods2019In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 345, p. 99-113Article in journal (Refereed)

We consider a multiscale strategy addressing the disparate scales in the Landau–Lifschitz equations in micromagnetism. At the microscopic scale, the dynamics of magnetic moments are driven by a high frequency field. On the macroscopic scale we are interested in simulating the dynamics of the magnetisation without fully resolving the microscopic scales.

The method follows the framework of heterogeneous multiscale methods and it has two main ingredients: a micro- and a macroscale model. The microscopic model is assumed to be known exactly whereas the macromodel is incomplete as it lacks effective quantities. The two models use different temporal and spatial scales and effective parameter values for the macromodel are computed on the fly, allowing for improved efficiency over traditional one-scale schemes.

For the analysis, we consider a single spin under a high frequency field and show that effective quantities can be obtained accurately with step-sizes much larger than the size of the microscopic scales required to resolve the microscopic features. Numerical results both for a single magnetic particle as well as a chain of interacting magnetic particles are given to validate the theory.

• 43.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. École Polytechniques Fédérale de Lausanne, Switzerland.
Uppsala University, Uppsala, Sweden.
An equation-free approach for second order multiscale hyperbolic problems in non-divergence form2018In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 16, no 8, p. 2317-2343Article in journal (Refereed)

The present study concerns the numerical homogenization of second order hyperbolic equations in non-divergence form, where the model problem includes a rapidly oscillating coefficient function. These small scales influence the large scale behavior, hence their effects should be accurately modelled in a numerical simulation. A direct numerical simulation is prohibitively expensive since a minimum of two points per wavelength are needed to resolve the small scales. A multiscale method, under the equation-free methodology, is proposed to approximate the coarse scale behaviour of the exact solution at a cost independent of the small scales in the problem. We prove convergence rates for the upscaled quantities in one as well as in multi-dimensional periodic settings. Moreover, numerical results in one and two dimensions are provided to support the theory.

• 44.
Uppsala University, Uppsala, Sweden; École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland.
Uppsala University, Uppsala, Sweden. Uppsala University, Uppsala, Sweden.
Atomistic-continuum multiscale modelling of magnetisation dynamics at non-zero temperature2018In: Advances in Computational Mathematics, ISSN 1019-7168, E-ISSN 1572-9044, Vol. 44, no 4, p. 1119-1151Article in journal (Refereed)

In this article, a few problems related to multiscale modelling of magnetic materials at finite temperatures and possible ways of solving these problems are discussed. The discussion is mainly centred around two established multiscale concepts: the partitioned domain and the upscaling-based methodologies. The major challenge for both multiscale methods is to capture the correct value of magnetisation length accurately, which is affected by a random temperature-dependent force. Moreover, general limitations of these multiscale techniques in application to spin systems are discussed.

• 45.
Ecole polytechnique f´ed´erale de Lausanne, Switzerland.
University of Warwick, Coventry, UK. Uppsala University, Sweden.
Modelling long-range interactions in multiscale simulations of ferromagnetic materials2020In: Advances in Computational Mathematics, ISSN 1019-7168, E-ISSN 1572-9044, Vol. 46, no 1, article id 2Article in journal (Other academic)

Atomistic-continuum multiscale modelling is becoming an increasingly popular tool for simulating the behaviour of materials due to its computational efficiency and reliable accuracy. In the case of ferromagnetic materials, the atomistic approach handles the dynamics of spin magnetic moments of individual atoms, while the continuum approximations operate with volume-averaged quantities, such as magnetisation. One of the challenges for multiscale models in relation to physics of ferromagnets is the existence of the long-range dipole-dipole interactions between spins. The aim of the present paper is to demonstrate a way of including these interactions into existing atomistic-continuum coupling methods based on the partitioned-domain and the upscaling strategies. This is achieved by modelling the demagnetising field exclusively at the continuum level and coupling it to both scales. Such an approach relies on the atomistic expression for the magnetisation field converging to the continuum expression when the interatomic spacing approaches zero, which is demonstrated in this paper.

• 46.
Uppsala University, Sweden.
KTH, Stockholm.
A time dependent approach for removing the cell boundary error in elliptic homogenization problems2016In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 314, p. 206-227Article in journal (Refereed)

This paper concerns the cell-boundary error present in multiscale algorithms for elliptic homogenization problems. Typical multiscale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. To solve the micro model, boundary conditions are required on the boundary of the microscopic domain. Imposing a naive boundary condition leads to O(ε/η) error in the computation, where ε is the size of the microscopic variations in the media and η is the size of the micro-domain. The removal of this error in modern multiscale algorithms still remains an important open problem. In this paper, we present a time-dependent approach which is general in terms of dimension. We provide a theorem which shows that we have arbitrarily high order convergence rates in terms of ε/η in the periodic setting. Additionally, we present numerical evidence showing that the method improves the O(ε/η) error to O(ε) in general non-periodic media.

• 47.
KTH, Stockholm.
KTH, Stockholm.
Analysis of heterogeneous multiscale methods for long time wave propagation problems2014In: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 12, no 3, p. 1135-1166Article in journal (Refereed)

In this paper, we analyze a multiscale method developed under the heterogeneous multiscale method (HMM) framework for numerical approximation of multiscale wave propagation problems in periodic media. In particular, we are interested in the long time $O(\varepsilon^{-2})$ wave propagation, where $\varepsilon$ represents the size of the microscopic variations in the media. In large time scales, the solutions of multiscale wave equations exhibit $O(1)$ dispersive effects which are not observed in short time scales. A typical HMM has two main components: a macromodel and a micromodel. The macromodel is incomplete and lacks a set of local data. In the setting of multiscale PDEs, one has to solve for the full oscillatory problem over local microscopic domains of size $\eta=O(\varepsilon)$ to upscale the parameter values which are missing in the macroscopic model. In this paper, we prove that if the microproblems are consistent with the macroscopic solutions, the HMM approximates the unknown parameter values in the macromodel up to any desired order of accuracy in terms of $\varepsilon/\eta$.

• 48.
Uppsala University, Sweden.
KTH, Stockholm.
Estimates for the Upscaling Error in Heterogeneous Multiscale Methods for Wave Propagation Problems in Locally Periodic Media2017In: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 15, no 2, p. 948-976Article in journal (Refereed)

This paper concerns the analysis of a multiscale method for wave propagation problems in microscopically nonhomogeneous media. A direct numerical approximation of such problems is prohibitively expensive as it requires resolving the microscopic variations over a much larger physical domain of interest. The heterogeneous multiscale method (HMM) is an efficient framework to approximate the solutions of multiscale problems. In the HMM, one assumes an incomplete macroscopic model which is coupled to a known but expensive microscopic model. The micromodel is solved only locally to upscale the parameter values which are missing in the macromodel. The resulting macroscopic model can then be solved at a cost independent of the small scales in the problem. In general, the accuracy of the HMM is related to how good the upscaling step approximates the right macroscopic quantities. The analysis of the method that we consider here was previously addressed only in purely periodic media, although the method itself is numerically shown to be applicable to more general settings. In the present study, we consider a more realistic setting by assuming a locally periodic medium where slow and fast variations are allowed at the same time. We then prove that the HMM captures the right macroscopic effects. The generality of the tools and ideas in the analysis allows us to establish convergence rates in a multidimensional setting. The theoretical findings here imply an improved convergence rate in one dimension, which also justifies the numerical observations from our earlier study.

• 49.
KTH, Sweden.
Laboratoire POEMS, UMA, ENSTA ParisTech, Palaiseau, France.
A finite element heterogeneous multiscale method with improved control over the modeling error2016In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 14, no 2, p. 463-487Article in journal (Refereed)

Multiscale partial differential equations (PDEs) are difficult to solve by traditional numerical methods due to the need to resolve the small wavelengths in the media over the entire computational domain. We develop and analyze a Finite Element Heterogeneous Multiscale Method (FE-HMM) for approximating the homogenized solutions of multiscale PDEs of elliptic, parabolic, and hyperbolic type. Typical multiscale methods require a coupling between a micro and a macro model. Inspired from the homogenization theory, traditional FE-HMM schemes use elliptic PDEs as the micro model. We use, however, the second order wave equation as our micro model independent of the type of the problem on the macro level. This allows us to control the modeling error originating from the coupling between the different scales. In a spatially fully discrete a priori error analysis we prove that the modeling error can be made arbitrarily small for periodic media, even if we do not know the exact period of the oscillations in the media. We provide numerical examples in one and two dimensions confirming the theoretical results. Further examples show that the method captures the effective solutions in general non-periodic settings as well.

• 50.
Shiraz Univ, Coll Sci, Dept Math, Shiraz, Iran..
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Shiraz Univ, Coll Sci, Dept Math, Shiraz, Iran..
Enveloping algebras of color hom-Lie algebras2019In: Turkish Journal of Mathematics, ISSN 1300-0098, E-ISSN 1303-6149, Vol. 43, no 1, p. 316-339Article in journal (Refereed)

In this paper, the universal enveloping algebra of color hom-Lie algebras is studied. A construction of the free involutive hom-associative color algebra on a hom-module is described and applied to obtain the universal enveloping algebra of an involutive hom-Lie color algebra. Finally, the construction is applied to obtain the well-known Poincare- Birkhoff-Witt theorem for Lie algebras to the enveloping algebra of an involutive color hom-Lie algebra.

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