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• 1. Aaghabali, M.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Upper bounds on the number of perfect matchings and directed 2-factors in graphs with given number of vertices and edges2015In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 45, p. 132-144Article in journal (Refereed)

We give an upper bound on the number of perfect matchings in simple graphs with a given number of vertices and edges. We apply this result to give an upper bound on the number of 2-factors in a directed complete bipartite balanced graph on 2n vertices. The upper bound is sharp for even n. For odd n we state a conjecture on a sharp upper bound.

• 2.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Optimering av kortaste vägen vid hantering och avledning av skadligt dagvatten: Lösning med A-stjärna algoritm samt en guide med ekonomiska styrmedel för beslutsfattande aktörer2017Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis

The earth's population is growing and increasingly more people move into urban areas. This means that as cities grow, new buildings are being built and infrastructures are expanding. This rapid growth is directly related to increased floods as a result of man-made changes in nature.

The already overloaded storm water systems for rain-, melt-, rinsing and other surplus water cannot often handle the existing demand. Therefore, floods arise at greater rain intensity and pose significant costs to society. Due to an unclear division of responsibility within the municipality's organizations there is a failure to handle the existing storm water problem. In order to be able to plan for sustainable cities in the future, it is important to find a viable solution regarding the responsibility issue and how to best handle the storm water to achieve cost advantage.

This study presents a guide for municipalities on how to allocate the responsibility between the municipality and the exploiter. The guide is based on simulations and theories in optimization to propose effective solutions for harmful surplus storm water. Through simulations of the storm water system, the amount of surplus water that does not fit the storm water system capacity has been quantified. In addition, to find a reasonable alternative run-off path for the surplus water, different methods of the shortest path problem have been investigated.

The results show that a classical shortest path algorithm with a heuristic function is not the most appropriate alternative. This because the heuristic function in the algorithm prevents the selection of a more natural pathway upstream even though it could be a more optimal solution.

• 3.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Numerical analysis for random processes and fields and related design problems2011Doctoral thesis, comprehensive summary (Other academic)

In this thesis, we study numerical analysis for random processes and fields. We investigate the behavior of the approximation accuracy for specific linear methods based on a finite number of observations. Furthermore, we propose techniques for optimizing performance of the methods for particular classes of random functions. The thesis consists of an introductory survey of the subject and related theory and four papers (A-D).

In paper A, we study a Hermite spline approximation of quadratic mean continuous and differentiable random processes with an isolated point singularity. We consider a piecewise polynomial approximation combining two different Hermite interpolation splines for the interval adjacent to the singularity point and for the remaining part. For locally stationary random processes, sequences of sampling designs eliminating asymptotically the effect of the singularity are constructed.

In Paper B, we focus on approximation of quadratic mean continuous real-valued random fields by a multivariate piecewise linear interpolator based on a finite number of observations placed on a hyperrectangular grid. We extend the concept of local stationarity to random fields and for the fields from this class, we provide an exact asymptotics for the approximation accuracy. Some asymptotic optimization results are also provided.

In Paper C, we investigate numerical approximation of integrals (quadrature) of random functions over the unit hypercube. We study the asymptotics of a stratified Monte Carlo quadrature based on a finite number of randomly chosen observations in strata generated by a hyperrectangular grid. For the locally stationary random fields (introduced in Paper B), we derive exact asymptotic results together with some optimization methods. Moreover, for a certain class of random functions with an isolated singularity, we construct a sequence of designs eliminating the effect of the singularity.

In Paper D, we consider a Monte Carlo pricing method for arithmetic Asian options. An estimator is constructed using a piecewise constant approximation of an underlying asset price process. For a wide class of Lévy market models, we provide upper bounds for the discretization error and the variance of the estimator. We construct an algorithm for accurate simulations with controlled discretization and Monte Carlo errors, andobtain the estimates of the option price with a predetermined accuracy at a given confidence level. Additionally, for the Black-Scholes model, we optimize the performance of the estimator by using a suitable variance reduction technique.

• 4.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Was it snowing on lake Kassjön in January 4486 BC? Functional data analysis of sediment data.2014In: Proceedings of the Third International Workshop on Functional and Operatorial Statistics (IWFOS 2014), Stresa, Italy, June 2014., 2014Conference paper (Refereed)
• 5.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Medicine, Department of Community Medicine and Rehabilitation, Physiotherapy. National Sports Institute of Malaysia. MOX – Department of Mathematics, Politecnico di Milano. Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics. Umeå University, Faculty of Medicine, Department of Community Medicine and Rehabilitation, Physiotherapy. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. MOX – Department of Mathematics, Politecnico di Milano.
An inferential framework for domain selection in functional anova2014In: Contributions in infinite-dimensional statistics and related topics / [ed] Bongiorno, E.G., Salinelli, E., Goia, A., Vieu, P, Esculapio , 2014Conference paper (Refereed)

We present a procedure for performing an ANOVA test on functional data, including pairwise group comparisons. in a Scheff´e-like perspective. The test is based on the Interval Testing Procedure, and it selects intervals where the groups significantly differ. The procedure is applied on the 3D kinematic motion of the knee joint collected during a functional task (one leg hop) performed by three groups of individuals.

• 6.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Multivariate piecewise linear interpolation of a random field2011Manuscript (preprint) (Other academic)

We consider a multivariate piecewise linear interpolation of a continuous random field on a-dimensional cube. The approximation performance is measured by the integrated mean square error. Multivariate piecewise linear interpolator is defined by N field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field in mean square sense and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields we provide the upper bound for the approximation accuracy in the uniform mean square norm.

• 7.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On the error of the Monte Carlo pricing method for Asian option2008In: Journal of Numerical and Applied Mathematics, ISSN 0868-6912, Vol. 96, no 1, p. 1-10Article in journal (Refereed)

We consider a Monte Carlo method to price a continuous arithmetic Asian option with a given precision. Piecewise constant approximation and plain simulation are used for a wide class of models based on L\'{e}vy processes. We give bounds of the possible discretization and simulation errors. The sufficient numbers of discretization points and simulations to obtain requested accuracy are derived. To demonstrate the general approach, the Black-Scholes model is studied in more detail. We undertake the case of continuous averaging and starting time zero,  but the obtained results can be applied to the discrete case  and generalized for any time before an execution date. Some numerical experiments and comparison to the PDE based method are also presented.

• 8.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Piecewise multilinear interpolation of a random field2013In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 45, no 4, p. 945-959Article in journal (Refereed)

We consider a piecewise-multilinear interpolation of a continuous random field on a d-dimensional cube. The approximation performance is measured using the integrated mean square error. Piecewise-multilinear interpolator is defined by N-field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field, in the mean square sense, and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields, we provide the upper bound for the approximation accuracy in the uniform mean square norm.

• 9.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Spline approximation of a random process with singularity2011In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171, Vol. 141, no 3, p. 1333-1342Article in journal (Refereed)

Let a continuous random process X defined on [0,1] be (m+β)-smooth, 0m, 0<β$\leq$1, in quadratic mean for all t>0 and have an isolated singularity point at t=0. In addition, let X be locally like a m-fold integrated β-fractional Brownian motion for all nonsingular points. We consider approximation of X by piecewise Hermite interpolation splines with n free knots (i.e., a sampling design, a mesh). The approximation performance is measured by mean errors (e.g., integrated or maximal quadratic mean errors). We construct a sequence of sampling designs with asymptotic approximation rate n^(m+β) for the whole interval.

• 10.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Stratified Monte Carlo quadrature for continuous random fields2015In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713, Vol. 17, no 1, p. 59-72Article in journal (Refereed)

We consider the problem of numerical approximation of integrals of random fields over a unit hypercube. We use a stratified Monte Carlo quadrature and measure the approximation performance by the mean squared error. The quadrature is defined by a finite number of stratified randomly chosen observations with the partition generated by a rectangular grid (or design). We study the class of locally stationary random fields whose local behavior is like a fractional Brownian field in the mean square sense and find the asymptotic approximation accuracy for a sequence of designs for large number of the observations. For the H¨older class of random functions, we provide an upper bound for the approximation error. Additionally, for a certain class of isotropic random functions with an isolated singularity at the origin, we construct a sequence of designs eliminating the effect of the singularity point.

• 11.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Politecnico di Milano, Italy. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Politecnico di Milano, Italy. Oslo University, Norway.
Clustering misaligned dependent curves applied to varved lake sediment for climate reconstruction2017In: Stochastic environmental research and risk assessment (Print), ISSN 1436-3240, E-ISSN 1436-3259, Vol. 31, no 1, p. 71-85Article in journal (Refereed)

In this paper we introduce a novel functional clustering method, the Bagging Voronoi K-Medoid Aligment (BVKMA) algorithm, which simultaneously clusters and aligns spatially dependent curves. It is a nonparametric statistical method that does not rely on distributional or dependency structure assumptions. The method is motivated by and applied to varved (annually laminated) sediment data from lake Kassjön in northern Sweden, aiming to infer on past environmental and climate changes. The resulting clusters and their time dynamics show great potential for seasonal climate interpretation, in particular for winter climate changes.

• 12.
Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The boundary Harnack inequality for variable exponent p-Laplacian, Carleson estimates, barrier functions and p(⋅)-harmonic measures2016In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 195, no 2, p. 623-658Article in journal (Refereed)

We investigate various boundary decay estimates for p(⋅)-harmonic functions. For domains in Rn,n≥2satisfying the ball condition (C1,1-domains), we show the boundary Harnack inequality for p(⋅)-harmonic functions under the assumption that the variable exponent p is a bounded Lipschitz function. The proof involves barrier functions and chaining arguments. Moreover, we prove a Carleson-type estimate for p(⋅)-harmonic functions in NTA domains in Rn and provide lower and upper growth estimates and a doubling property for a p(⋅)-harmonic measure.

• 13.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Predictive Modeling of Emissions: Heavy Duty Vehicles2016Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
• 14.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Analys av risker med garantinivåer i förhållande till förväntade utbetalningar och portföljavkastningar för traditionella pensionsförsäkringar: Ett examensarbete för Folksam Liv med dotterbolag2017Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
• 15.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Lekande lätt: att lära matematik utomhus på ett sociokulturellt sätt.2007Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis

Tanken bakom detta examensarbete var att framställa ett laborativt läromedel i matematik för utomhusmiljö som ska upplevas som motivationshöjande och lustfyllt för eleverna. Läromedlet är upplagt i lektionsplaneringar som är förankrade i läroplanen för det obligatoriska skolväsendet (1994), i kursplanen för matematik samt i de sex aspekterna på lärande ur sociokulturellt perspektiv som Dysthe (2003) skriver om. Vi har arbetat fram ett material som innefattar rumsuppfattning och mätning eftersom dessa passar utmärkt att genomföra i utemiljö. Idéerna till lektionsplaneringarna är utifrån oss själva men inspiration från tidigare kurser, kurslitteratur och VFU-platser går inte att frånse. Upplägget på lektionerna är utifrån Lindström och Pennlerts (2003) modell, där flera didaktiska frågeställningar tas i beaktning. Resultatet på detta examensarbete är de tio lektionsplaneringar som vi arbetat fram samt de kopplingar som vi har sett till de sex aspekterna och utomhusmiljön. Slutsatsen är en bekräftelse för oss och våra teorier om att undervisning utomhus ur ett sociokulturellt perspektiv går att förverkliga i lektionsplaneringar. Vi hoppas att flera lärare kommer att utnyttja vårt material för att få variation i sin undervisning.

• 16.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Partiformning vid intern materialförsörjning och layoutanpassning av lager: En fallstudie vid GE Healthcare Umeå av två-binge, supermarkets ochmaterialspindlar2014Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis

GE Healthcare (GEHC) Umeå har vid sin implementering av Lean genomfört förändringar i lagerstrukturen som är i behov av bättre anpassning. GEHC Umeås nya lagerstruktur innebär öppna supermarkets med förändrade försörjningsrutiner från lager till produktion. Ett två-binge system har implementeras där signalbehållare med material fylls av lageransvariga materialspindlar.

Det första identifierade problemet och forskningsfrågan utgör två-bingarnas kvantiteter som bestämmer mängden artiklar vid monteringsstationerna. Dessa behöver ses över och en rutin för bestämmande av kvantitet behöver etableras. Som den andra, och oberoende forskningsfrågan, har antalet supermarkets (lager) och dessmaterialspindlar identifieras som är många till antalet och utspridda med begränsad samordning.

Ett tidigare examensarbete, litteraturstudier, intervjuer samt egna observationer på plats har används för att beskriva nuläget genom både kvalitativa och kvantitativa metoder. På grund av bristen på liknande problem i litteraturen har externa partiformningsmetoder och lagerstyrning används och komplimenterats med simulering för två-binge systemet som del i besvarandet av den första frågeställningen. För den andra frågeställningen har delar ur förenklad systematisk lokalplanläggning används där bland annat olika centraliseringsgrader undersökts med simulering av materialtransporter vid olika artikelplaceringar.

Idag sätts kvantiteten efter prognoserat användande utifrån personliga erfarenheter. Samordningen mellan materialspindlarna är bristande och nyttjandegraden upplevs ojämn samtidigt som godsmottagningen skulle gynnas av ökad kapacitet. Standardiserade processer i materialhanteringen saknas och produktionsgrupperna har skilda arbetssätt som antaskunna gynnas av en centralisering där gemensamma rutiner lättare kan etableras.

De historiska transaktionerna visar att det finns utrymme för förbättringar då vissa artiklar genererar långa transportsträckor på grund lagerplats i förhållande till var de används iproduktionen. De nya binge-kvantiteterna från partiformningsmetoderna EOQ, m-EOQ och Kanbanformeln har testats i simulering av påfyllning och materialåtgång via en implementation i Excel VBA.

Kanbanformeln uppvisar högsta servicenivån 90 %, för lägsta totalkostnaden och minskad kapitalbindning. Kanbankvantiteterna minskar den totala kostnaden med 20 %. Antalet påfyllningar skulle öka med 7 % och antalet artiklar i produktion minskar med 59%. För layoutanpassningen har även simulering av olika orderplock och artikelplaceringen genomförts. Resultatet visar att en centraliseringsgrad är möjlig med en liten ökning avmaterialtransporterna. Det framgår även att artiklar som plockas väldigt sällan är beräknade att ta upp 89 hyllställage av totalt 230 stycken och bör ses över. Detta tillsammans med kravspecifikationen från analys-delen har hjälpt för att generera olika koncept.

GEHC Umeå bör använda Kanbanformeln i framtiden för bestämmande av kvantiteten i bingarna. Vissa anpassningar för gemensamma artiklar i Comm-lagret och artiklar utan historiska efterfrågan bör ske. För layouten bör GEHC Umeå först och främst flytta artiklarsom idag bidrar med onödiga transporter. På längre sikt bör en ökad grad centralisering avlagren vara möjlig med hänsyn till fördelar vid samordning och informell spridning av arbetsrutiner. Materialspindlarna bör underlätta för godsmottagningen, delta i bristrapportering samt förbättringsarbetet. Utöver detta bör möjligheter till ökat samarbetet mellan materialplaneringen, produktionsplaneringen och materialspindlarna undersökas

• 17. Aidanpää, Jan - Olov
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Developments in rotor dynamical modeling of hydropower units2009Conference paper (Refereed)
• 18.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Resonemang och problemlösning i Matematik 1b: En uppgiftsanalys av två läroböcker2018Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis

Studien undersökte möjligheten att utveckla problemlösnings- och resonemangsförmåga i svenska läroböcker. Under studiens gång genomfördes en läromedelsanalys av två läroböcker för gymnasiekursen Matematik 1b, Matematik 5000 och Matematik Origo. Analysen tog hänsyn till olika aspekter av förmågorna, olika svårighetsgrader på uppgifterna samt olika centralt innehåll i ämnesplanen. Av 2223 uppgifter i Matematik 5000 kunde problemlösningsförmågan utvecklas i 5% och resonemangsförmågan i 6%. Motsvarande andelar i 1747 uppgifter i Matematik Origo var 7% respektive 10%. Analysen visade också att större andel av uppgifterna på de två högre svårighetsnivåerna gav möjlighet att utveckla förmågorna än de på den lägsta nivån i båda böckerna. Det framkom även att inte alla aspekter av förmågorna fick lika stort utrymme att utvecklas. I båda böckerna var det geometri-kapitlet som fokuserade på dessa förmågor i störst andel av uppgifterna.

• 19. Akbari, Saieed
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On 1-sum flows in undirected graphs2016In: The Electronic Journal of Linear Algebra, ISSN 1537-9582, E-ISSN 1081-3810, Vol. 31, p. 646-665Article in journal (Refereed)

Let G = (V, E) be a simple undirected graph. For a given set L subset of R, a function omega: E -> L is called an L-flow. Given a vector gamma is an element of R-V , omega is a gamma-L-flow if for each v is an element of V, the sum of the values on the edges incident to v is gamma(v). If gamma(v) = c, for all v is an element of V, then the gamma-L-flow is called a c-sum L-flow. In this paper, the existence of gamma-L-flows for various choices of sets L of real numbers is studied, with an emphasis on 1-sum flows. Let L be a subset of real numbers containing 0 and denote L* := L \ {0}. Answering a question from [S. Akbari, M. Kano, and S. Zare. A generalization of 0-sum flows in graphs. Linear Algebra Appl., 438:3629-3634, 2013.], the bipartite graphs which admit a 1-sum R* -flow or a 1-sum Z* -flow are characterized. It is also shown that every k-regular graph, with k either odd or congruent to 2 modulo 4, admits a 1-sum {-1, 0, 1}-flow.

• 20.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
A Comparison of Tests for Ordered Alternatives With Application in Medicine1997Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

A situation frequently encountered in medical studies is the comparison of several treatments with a control. The problem is to determine whether or not a test drug has a desirable medical effect and/or to identify the minimum effective dose. In this Bachelor’s thesis, some of the methods used for testing hypotheses of ordered alternatives are reviewed and compared with respect to the power of the tests. Examples of multiple comparison procedures, maximum likelihood procedures, rank tests and different types of contrasts are presented and the properties of the methods are explored.

Depending on the degree of knowledge about the dose-responses, the aim of the study, whether the test is parametric or non-parametric and distribution-free or not, different recommendations are given which of the tests should be used. Thus, there is no single test which can be applied in all experimental situations for testing all different alternative hypotheses.

• 21.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, Department of Computing Science.
How will your workload look like in 6 years?: Analyzing Wikimedia's workload2014In: Proceedings of the 2014 IEEE International Conference on Cloud Engineering (IC2E 2014) / [ed] Lisa O’Conner, IEEE Computer Society, 2014, p. 349-354Conference paper (Refereed)

Accurate understanding of workloads is key to efficient cloud resource management as well as to the design of large-scale applications. We analyze and model the workload of Wikipedia, one of the world's largest web sites. With descriptive statistics, time-series analysis, and polynomial splines, we study the trend and seasonality of the workload, its evolution over the years, and also investigate patterns in page popularity. Our results indicate that the workload is highly predictable with a strong seasonality. Our short term prediction algorithm is able to predict the workload with a Mean Absolute Percentage Error of around 2%.

• 22.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, Department of Computing Science.
Measuring cloud workload burstiness2014In: 2014 IEEE/ACM 7th International Conference on Utility and Cloud Computing (UCC), IEEE conference proceedings, 2014, p. 566-572Conference paper (Refereed)

Workload burstiness and spikes are among the main reasons for service disruptions and decrease in the Quality-of-Service (QoS) of online services. They are hurdles that complicate autonomic resource management of datacenters. In this paper, we review the state-of-the-art in online identification of workload spikes and quantifying burstiness. The applicability of some of the proposed techniques is examined for Cloud systems where various workloads are co-hosted on the same platform. We discuss Sample Entropy (SampEn), a measure used in biomedical signal analysis, as a potential measure for burstiness. A modification to the original measure is introduced to make it more suitable for Cloud workloads.

• 23.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Where to Stack the Chocolate?: Mapping and Optimisation of the Storage Locations with Associated Transportation Cost at Marabou2017Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis

Today, inventory management at Marabou is organised in such way that articles are stored based on which production line they belong to and are sent to storage locations close to their production line. However, some storage locations are not optimised, insofar articles are stored out of pure habit and follow what is considered most convenient. This means that the storage locations are not based on any fixed instructions or standard. In this report, we propose optimal storage locations with respect to transportation cost by modelling the problem mathematically as a minimal cost matching problem, which we solve using the so-called Hungarian algorithm. To be able to implement the Hungarian algorithm, we collected data regarding the stock levels of articles in the factory throughout 2016. We adjusted the collected data by turning the articles into units of pallets. We considered three different implementations of the Hungarian algorithm. The results from the different approaches are presented together with several suggestions regarding pallet optimisation. In addition to the theoretical background, our work is based on an empirical study through participant observations as well as qualitative interviews with factory employees. In addition to our modelling work, we thus offer several further suggestions for efficiency savings or improvements at the factory, as well as for further work building on this report.

Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Random iteration of isometries in unbounded metric spaces2003In: Nonlinearity, Vol. 16, p. 1107-1117Article in journal (Refereed)
• 25.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Läroboksbaserad eller lärobokslös matematikundervisning?: En inblick i hur lärare i grundskolans tidigare år resonerar vid val av undervisningssätt2009Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis

Syftet med undersökningen är att få en inblick i hur lärare i grundskolans tidigare år resonerar vid valet att undervisa läroboksbaserat eller lärobokslöst i matematik. Undersökningen riktar sig till lärarstudenter och verksamma lärare som arbetar med matematikämnet. Matematik är ett aktuellt ämne i dagens utbildningsdebatt, framförallt angående lärobokens vara eller icke vara i undervisningen. Frågeställningarna i denna undersökning berör vilka faktorer, och hur dessa faktorer, påverkar lärarnas val att undervisa läroboksbaserat eller lärobokslöst. Hur påverkar nämnda faktorer arbetssättet, samt vilka likheter och skillnader finns det mellan läroboksbaserad och lärobokslös matematikundervisning. Kvalitativa intervjuer genomfördes med sex olika lärare varav tre arbetar läroboksbaserat och tre lärobokslöst. Resultatet visar att likheterna är större än skillnaderna mellan dessa två arbetssätt då alla sex lärarna använder sig av läroböcker och laborativa material. Den stora skillnaden är användandets utgångspunkt.

• 26.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
A Risk and Capital Requirement Model for Life Insurance Portfolios2008Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

The capital requirements for insurance companies in the Solvency I framework are based on the premium and claim expenditure. This approach does not take the individual risk of the insurer into consideration and give policy holder little assur- ance. Therefore a framework called Solvency II is under development by EU and its members. The capital requirements in Solvency II are based on risk management and is related to the specific risks of the insurer. Moreover, the insurer must make disclosures both to the supervising authority and to the market. This puts pressure on the insurance companies to use better risk and capital management, which gives the policy holders better assurance.

In this thesis we present a stochastic model that describes the development of assets and liabilities. We consider the following risks: Stock market, bond market, interest rate and mortality intensity. These risks are modeled by stochastic processes that are aggregated to describe the change in the insurers Risk Bearing Capital. The capital requirement, Solvency Capital Requirement, is calculated using Conditional Value-at-Risk at a 99% confidence level and Monte Carlo simulation. The results from this model is compared to the Swiss Solvency Test model for three different types of life insurance policies. We can conclude that for large portfolios, the model presented in this thesis gives a lower solvency capital requirement than the Swiss model for all three policies. For small portfolios, the capital requirement is larger due to the stochastic mortality risk which is not included in the Swiss model.

• 27.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Modellering av säkringsstrategier för en elförsäljningsportfölj2015Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis

Because of the high volatility of the electricity spot price there is a necessity of hedging the sales price of the production. The electricity spot price are volatile and are affected by climate, producer supply and political decisions. This means that the revenues from the power activites can vary alot from one year to another. The revenues from the power activites are especially important to be able to budget with probability since they are included in the total budget of Umeå municipality. In order to evaluate possible investment strategies the production along with the electricity spot- and futures prices of different maturities are modelled together. The modelling of production and electricity spot prices are based on a general seasonal block bootstrap method. Furthermore, two different assumptions are made about the relationship between spot prices and futures prices. The first emprical model is based on an assumption that there exists a mismatch between spot prices and the futures prices and historical differences are used to calculate this. The second model is based on the assumption that the futures prices are the same as the expected future spot price and that these are consistent.

The municipality’s current strategy is to hedge 300 GWh of the annual production in futures with three different maturities and sell the remaining of the production at spot price. This strategy can be seen as an average of four electricity prices and therefore reduces the risk of mismatch between the futures and spot price.

The empirical study show that historically it has been most profitable to invest in futures with maturity of three years. This has to do with the historical differences between futures prices and the electricity spot prices for this maturity has been the largest and thus gives the highest expected sales profit in the model. Furthermore, the study show that it has been more profitable to invest in futures compared with selling to spot price. Whether this is something that will continue into the future is uncertain due to the nature of the futures contract and the pricing of these. Finally, the study also show which investment strategies has been most profitable in a so-called backtest.

• 28.
Umeå University, Faculty of Medicine, Clinical Microbiology, Clinical Bacteriology.
Umeå University, Faculty of Medicine, Clinical Microbiology, Clinical Bacteriology. Umeå University, Faculty of Medicine, Clinical Microbiology, Clinical Bacteriology. Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics. FOI, Umeå (Swedish Defence Research Agency). Umeå University, Faculty of Medicine, Clinical Microbiology, Clinical Bacteriology.
T A microarray analysis of the murine macrophage response to infection with Francisella tularensis LVS2006In: Journal of Medical Microbiology, ISSN 0022-2615, E-ISSN 1473-5644, Vol. 55, no 8, p. 1023-1033Article in journal (Refereed)
• 29.
Umeå University, Faculty of Medicine, Department of Clinical Microbiology, Clinical Bacteriology.
Umeå University, Faculty of Medicine, Department of Clinical Microbiology, Clinical Bacteriology. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. NRC, Kanada. Umeå University, Faculty of Medicine, Department of Clinical Microbiology, Clinical Bacteriology.
Transcriptional profiling of the host responses in mouse lungs following aerosol2006In: Journal of Medical Microbiology, ISSN 0022-2615, E-ISSN 1473-5644, Vol. 55, no 3, p. 263-271Article in journal (Refereed)
• 30.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Effects of Physiological Variations2006Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

Heart ischemia, the precursor to an infarction, is one of the most common diseases in the western world. Today, the electrocardiogram or ECG is the most widely used tool to diagnose the disease. However, it often fails to detect the ischemia or to give an adequate picture of the size and location.

Therefore, the potential of increasing knowledge obtained through mathe- matical models is very high. In this thesis the bidomain model is used to describe the electrical activity in the heart and body with ischemia incorporated into the model. To solve the equations set up by the bidomain model, the finite element method is used. Different physiological variations have been made to the body, these include changing the location of the heart and varying the conductivities in the body. The solution to the equations is then studied at the body surface. The main question asked is whether it is possible to detect the location and size of different types of ischemia by analyzing the solution.

The methods used for this have been Singular Value Decomposition and Su- pervised learning. The different vectors obtained from the decomposition are used to distinguish the location and size of the ischemia for different physiolog- ical variations.

The results show that it is possible to distinguish the location of the ischemia but that it probably will be more difficult to find the correct size since the change in size is harder to separate from other physiological variations, such as the conductivity of the body.

Although relatively simple methods have been used, they indicate that, with further development, they can be used for the purpose of detecting the different ischemia.

• 31.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Likhetstecknet: Hur verksamma lärare introducerar och arbetar med förståelsen för likhetstecknet2014Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis

Syftet med mitt examensarbete är att utifrån de didaktiska frågorna, vad, hur och varför, undersöka hur verksamma lärare introducerar och arbetar med förståelsen för likhetstecknet i skolår 1. I litteraturgenomgången går jag igenom abstrakta symboler, styrdokument, Malmers inlärningsnivåer, Vygotsky och likhetstecknet. Undersökningen är gjord på två mellanstora skolor med ca 60 elever och en liten skola med endast 10 elever, samtliga skolor ligger i mellersta Sverige.

Jag har använt mig av en kvalitativ metod för min studie där jag gjort intervjuer med sammanlagt fem lärare. Grundfrågorna för intervjun är hur lärarna introducerar likhetstecknet i matematikundervisningen, vad de använder för material och uppgifter, hur de kontrollerar elevernas förståelse samt hur arbetet förändrats med tiden. Alla de intervjuade lärarna introducerar likhetstecknet i matematikundervisningen i början av höstterminen när eleverna just börjat skolan. De material och uppgifter som lärarna använder vid introduktion av likhetstecknet varierar mellan klossar, counters, pengar och vatten i samband med olika praktiska uppgifter och samtal. Ett material som samtliga lärare använder är balansvågen som visar praktiskt på ett tydligt sätt hur det ska vara lika mycket på båda sidor. Samtliga av de intervjuade lärarna använder matematikbok men det varierar mellan vilken bok som används.

Flera av lärarna berättar om att de får en känsla av om eleverna fått förståelse för likhetstecknet.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Explicit representation of membership in polynomial ideals2011In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 349, no 2, p. 345-365Article in journal (Refereed)

We introduce a new division formula on projective space which provides explicit solutions to various polynomial division problems with sharp degree estimates. We consider simple examples as the classical Macaulay theorem as well as a quite recent result by Hickel, related to the effective Nullstellensatz. We also obtain a related result that generalizes Max Noether's classical AF + BG theorem.

• 33.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
"Jag vet detta men kan inte förklara": En studie av gymnasieelevers förmåga att kommunicera  innebörden av grundläggande matematiska begrepp2009Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis

I skolans styrdokument står det tydligt att eleverna ska utveckla sin förmåga att kommunicera matematik samt använda lämpliga och korrekta begrepp. Syftet med examensarbetet är att undersöka om gymnasieskolans elever har förståelse gällande matematiska grundbegrepp med avseende på kommunikativ och funktionell förståelse samt se om det finns några skillnader mellan dessa. Med kommunikativ förståelse menas om eleverna kan förklara begrepp med egna ord och/eller med hjälp av figurer. Med funktionell förståelse menas om eleverna kan lösa uppgifter där olika begrepp står i fokus. För att undersöka detta valdes nio olika klasser ut och de fick vid olika tillfällen genomföra två prov som testade 12 grundläggande begrepp. På det första provet skulle eleverna med egna ord eller figurer förklara de 12 begreppen. På det andra provet skulle eleverna lösa 12 olika uppgifter som var och en innehöll ett specifikt begrepp. Undersökningen tyder på att eleverna har problem med den kommunikativa förståelsen då de endast gav en korrekt förklaring på ungefär hälften av begreppen. Däremot kan elever lösa uppgifter med begreppet i fokus vilket visar att eleverna har en större funktionell förståelse än de har kommunikativ förståelse.

• 34.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
”Jag räknar lite med huvudet och lite med händerna”: en studie om barns tankar kring fenomenet addition2007Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis

Syftet med vårt examensarbete är att genom barnsamtal samt observationer, studera vilka räknestrategier barn i år ett använder sig av i mötet med fenomenet addition. Det är dock inte endast fenomenet addition som är det intressanta, utan även vägen fram till en bättre förståelse för barnens tankar, detta för att vi ska vara bättre rustade i vår framtida undervisning. Då alla ser världen på skilda sätt, kommer barnen till skolan med olika förförståelse. Av den anledningen är det viktigt att pedagogen besitter verktyg, vilka kan hjälpa det enskilda barnet att utveckla sina kunskaper på bästa sätt. Frågeställningen vi avsett att undersöka är: vilka räknestrategier använder sig elever i år ett av i mötet med additionsuppgifter inom talområdet 1- 10? En viktig slutsats vi kom fram till under vårt arbete är att eleverna använder sig av olika räknestrategier samt att de har olika förkunskaper. Av denna anledning är det betydelsefullt att undervisningen individanpassas. En annan slutsats är att barnsamtal och observationer kan vara bra sätt för pedagogen att uppmärksamma och följa elevernas utveckling. Med hjälp av dessa metoder kan elevernas kunskaper kartläggas och eleven blir medveten om sitt eget lärande.

• 35.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Modelling Discrete Data for Control Chart Application: A Quality Improvement Project at Volvo GTO Umeå2017Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis

In any manufacturing process, it is of vital importance to improve and maintain a high quality outcome. One way of doing so is to make use of statistical process control (SPC), which is a collection of tools for monitoring the outcome of a process. The aim of this work has been to investigate SPC methods suitable for modelling discrete data describing quality, as well as implementing these methods on quality data of cabs collected at the paint shop of Volvo GTO Umeå. A tool that has been of special importance in this project is control charts, and a great part of the project has consisted of finding suitable statistical distributions on which to base these charts. Methods for this include goodness-of-fit of distributions, as well as regression based on generalized linear models (GLM), for finding suitable distributions and estimate their parameters. The regression models also provided useful information on how the quality data, that describes counts of defects in the paint, depend on background variables of a production unit. As a rule, the superiority of one regression model or distribution over another has been evaluated with the Akaike Information Criterion (AIC).

The results of this project are the GLM models that showed the highest significance, as well as implementations of simple Shewhart type control charts with different control limits corresponding to the parameter estimates of these models. The control limits proved to differ from each other for different observations, depending on which underlying combination on values on background variables, such as cab type, that the observation had. In general, there were also indications that the negative binomial distribution works well for modelling relatively common defect types, or sums of counts of different defect types, whereas zero-inflated might be better for less common defect types.

• 36.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Matematikindividualisering: - hur två lärare ser på det2005Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis

I mitt examensarbete har jag valt att undersöka matematikindividualisering lite närmare. Mitt syfte var att ta reda på hur två lärare ser på detta och om individualiseringen kommer till uttryck i deras undervisning och i så fall hur. Jag har använt mig av en kvalitativ metod, i det här fallet observationer och intervjuer. Observationerna utfördes i två klasser och intervjuerna gjordes med lärarna för dessa klasser. Genom att använda mig av de här metoderna så hoppades jag få en lite djupare kunskap om hur lärarna i min undersökning tänkte och arbetade. I min bakgrund har jag tagit upp sådan information som jag anser att man bör känna till som läsare, framför allt om man inte känner till så mycket om individualisering sedan tidigare. De slutsatser jag har dragit är att de båda lärarna individualiserar till en viss del, men att de tycker att det är svårt. Vidare har jag kommit fram till att man för att kunna arbeta individualiserat måste ha både ämneskunskaper och didaktiska kunskaper för att få det att fungera tillfredsställande.

• 37.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Fairness and Flexibility in Oral Examination2005Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis

This is a descriptive ethnographical study with the purpose of examining teachers’ and students’ experiences of oral examination at a State Pedagogical University in western Russia. The study also focused on finding the characteristics of oral examination and the contextual factors influencing its implementation. The research was done using participatory observations and interviews. The results show that interviewees experience oral assessment in general as positive. Their descriptions are summarised and analysed using a number of key concepts, of which flexibility, subjectivity, individualisation, and fairness are the most important. The study also shows that contextual factors such as culture, traditions, and organisational framework have large impact on how the examination is done. The conclusion is that oral examination has both gins and losses, since the teacher’s active participation creates possibilities for individualisation and deep probing of the students’ knowledge, but is also a source of bias because of its subjectivity.

• 38.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
On the complexity of matrix reduction over finite fields2006Report (Other academic)
• 39.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The bivariate ising polynomial of a graph2009In: Discrete Applied Mathematics, ISSN 0166-218X, E-ISSN 1872-6771, Vol. 157, no 11, p. 2515-2524Article in journal (Refereed)

In this paper we discuss the two variable Ising polynomials in a graph theoretical setting. This polynomial has its origin in physics as the partition function of the Ising model with an external field. We prove some basic properties of the Ising polynomial and demonstrate that it encodes a large amount of combinatorial information about a graph. We also give examples which prove that certain properties, such as the chromatic number, are not determined by the Ising polynomial. Finally we prove that there exist large families of non-isomorphic planar triangulations with identical Ising polynomial. (C) 2009 Published by Elsevier B.V.

• 40.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Avoiding Arrays of Odd Order by Latin Squares2013In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 22, no 2, p. 184-212Article in journal (Refereed)

We prove that there is a constant c such that, for each positive integer k, every (2k + 1) x (2k + 1) array A on the symbols 1, ... , 2k + 1 with at most c(2k + 1) symbols in every cell, and each symbol repeated at most c(2k + 1) times in every row and column is avoidable; that is, there is a (2k + 1) x (2k + 1) Latin square S on the symbols 1, ... , 2k + 1 such that, for each i, j is an element of {1, ... , 2k + 1}, the symbol in position (i, j) of S does not appear in the corresponding cell in Lambda. This settles the last open case of a conjecture by Haggkvist. Using this result, we also show that there is a constant rho, such that, for any positive integer n, if each cell in an n x n array B is assigned a set of m <= rho n symbols, where each set is chosen independently and uniformly at random from {1, ... , n}, then the probability that B is avoidable tends to 1 as n -> infinity.

• 41.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On the Ising problem and some matrix operations2007Doctoral thesis, comprehensive summary (Other academic)

The first part of the dissertation concerns the Ising problem proposed to Ernst Ising by his supervisor Wilhelm Lenz in the early 20s. The Ising model, or perhaps more correctly the Lenz-Ising model, tries to capture the behaviour of phase transitions, i.e. how local rules of engagement can produce large scale behaviour.

Two decades later Lars Onsager solved the Ising problem for the quadratic lattice without an outer field. Using his ideas solutions for other lattices in two dimensions have been constructed. We describe a method for calculating the Ising partition function for immense square grids, up to linear order 320 (i.e. 102400 vertices).

In three dimensions however only a few results are known. One of the most important unanswered questions is at which temperature the Ising model has its phase transition. In this dissertation it is shown that an upper bound for the critical coupling Kc, the inverse absolute temperature, is 0.29 for the tree dimensional cubic lattice.

To be able to get more information one has to use different statistical methods. We describe one sampling method that can use simple state generation like the Metropolis algorithm for large lattices. We also discuss how to reconstruct the entropy from the model, in order to obtain parameters as the free energy.

The Ising model gives a partition function associated with all finite graphs. In this dissertation we show that a number of interesting graph invariants can be calculated from the coefficients of the Ising partition function. We also give some interesting observations about the partition function in general and show that there are, for any N, N non-isomorphic graphs with the same Ising partition function.

The second part of the dissertation is about matrix operations. We consider the problem of multiplying them when the entries are elements in a finite semiring or in an additively finitely generated semiring. We describe a method that uses O(n3 / log n) arithmetic operations.

We also consider the problem of reducing n x n matrices over a finite field of size q using O(n2 / logq n) row operations in the worst case.

• 42.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Series expansion for the density of states of the Ising and Potts modelsManuscript (Other academic)
• 43.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
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)

In this paper we show that n×n matrices with entries from a semiring R which is generated additively by q generators can be multiplied in time O(q2nω), where nω is the complexity for matrix multiplication over a ring (Strassen: ω<2.807, Coppersmith and Winograd: ω<2.376).

We first present a combinatorial matrix multiplication algorithm for the case of semirings with q elements, with complexity O(n3/log2qn), matching the best known methods in this class.

Next we show how the ideas used can be combined with those of the fastest known boolean matrix multiplication algorithms to give an O(q2nω) algorithm for matrices of, not necessarily finite, semirings with q additive generators.

For finite semirings our combinatorial algorithm is simple enough to be a practical algorithm and is expected to be faster than the O(q2nω) algorithm for matrices of practically relevant sizes.

• 44.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
On the complexity of matrix reduction over finite fields2007In: Advances in Applied Mathematics, ISSN 0196-8858, Vol. 39, no 4, p. 428-452Article in journal (Refereed)

We study matrix elimination over finite fields, and present an algorithm which is asymptotically faster than the traditional Gauss--Jordan elimination. We also bound the average and worst-case complexity for the problem, proving that our algorithm is close to being optimal, and show related concentration results for random matrices.

Next we present the results of a large computational study of the complexities for small matrices and fields. Here we determine the exact distribution of the complexity for matrices from $\mathrm{GL}_{n}(\mathbb{F}_{q})$, with $n$ an $q$ small

Finally we consider an extension of the problems studied for finite fields to finite semifields. We give a conjecture on the behaviour of a natural analogue of $\mathrm{GL}_{n}$ for semifields and prove this for a certain class of semifields.

• 45.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Avoidability by Latin squares of arrays of even orderManuscript (preprint) (Other academic)

We prove that for any k and any 2k × 2k array A such that no cell in A contains more than   k/2550 symbols, and no symbol occurs more than k/2550 times in any row or column, there is a Latin square such that no 2550cell in the Latin square contains a symbol that occurs in the corresponding cell in A. This proves a conjecture of Häggkvist [8] in the special case of arrays with even side.

• 46.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Avoidability of random arraysManuscript (preprint) (Other academic)

An n×n array that in each cell contains a subset of the symbols 1, . . . , n is avoidable if there exists a Latin square of order n such that no cell in the Latin square contains a symbol which belongs to the set of symbols in the corresponding cell of the array. Some results on deterministic conditions for avoidability of arrays have been found, but here we study the problem of having an array with randomly assigned subsets of C in its cells. This is equivalent to the problem of list-edge-coloring $K_{n,n}$ with randomly assigned lists from the set {1, . . . , n}. We show that an array where each symbol appears in each cell with probability p will be avoidable with very high probability even if p is such that the expected number of symbols forbidden in each cell is slightly higher than what deterministic theorems can prove is avoidable.

• 47.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Avoiding (m, m, m)-arrays of order n = 2kManuscript (preprint) (Other academic)

An (m, m, m)-array of order n is an n × n array such that each cell is assigned a set of at most m symbols from {1,...,n} such that no symbol occurs more than m times in any row or column. An (m,m,m)- array is called avoidable if there exists a Latin square such that no cell in the Latin square contains a symbol that also belongs to the set assigned to the corresponding cell in the array. We show that there is a constant γ such that if m ≤ γ2k, then any (m,m,m)-array of order 2k is avoidable. Such a constant γ has been conjectured to exist for all n by Häggkvist.

• 48.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Avoiding (m, m, m)-arrays of order n=2(k)2012In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 19, no 1, p. P63-Article in journal (Refereed)

An (m, m, m)-array of order n is an n x n array such that each cell is assigned a set of at most m symbols from f 1,...,n g such that no symbol occurs more than m times in any row or column. An (m, m, m)-array is called avoidable if there exists a Latin square such that no cell in the Latin square contains a symbol that also belongs to the set assigned to the corresponding cell in the array. We show that there is a constant gamma such that if m <= gamma 2(k) and k >= 14, then any (m, m, m)-array of order n = 2(k) is avoidable. Such a constant gamma has been conjectured to exist for all n by Haggkvist.

• 49.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On Latin squares and avoidable arrays2010Doctoral thesis, comprehensive summary (Other academic)

This thesis consists of the four papers listed below and a survey of the research area.

I Lina J. Andrén: Avoiding (m, m, m)-arrays of order n = 2k

II Lina J. Andrén: Avoidability of random arrays

III Lina J. Andr´en: Avoidability by Latin squares of arrays with even order

IV Lina J. Andrén, Carl Johan Casselgren and Lars-Daniel Öhman: Avoiding arrays of odd order by Latin squares

Papers I, III and IV are all concerned with a conjecture by Häggkvist saying that there is a constant c such that for any positive integer n, if m ≤ cn, then for every n × n array A of subsets of {1, . . . , n} such that no cell contains a set of size greater than m, and none of the elements 1, . . . , n belongs to more than m of the sets in any row or any column of A, there is a Latin square L on the symbols 1, . . . , n such that there is no cell in L that contains a symbol that belongs to the set in the corresponding cell of A. Such a Latin square is said to avoid A. In Paper I, the conjecture is proved in the special case of order n = 2k . Paper III improves on the techniques of Paper I, expanding the proof to cover all arrays of even order. Finally, in Paper IV, similar methods are used together with a recoloring theorem to prove the conjecture for all orders. Paper II considers another aspect of the problem by asking to what extent way a deterministic result concerning the existence of Latin squares that avoid certain arrays can be used when the sets in the array are assigned randomly.

• 50.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Avoiding arrays of odd order by Latin squaresManuscript (preprint) (Other academic)

We prove that there exists a constant c such that for each pos- itive integer k every (2k+1)×(2k+1) array A on the symbols 1,...,2k+1 with at most c(2k + 1) symbols in every cell, and each symbol repeated at most c(2k+1) times in every row and column is avoidable; that is, there is a (2k+1)×(2k+1) Latin square S on the symbols 1,...,2k+1 such that for each cell (i, j) in S the symbol in (i, j) does not appear in the corresponding cell in A. This settles the last open case of a conjecture by Häggkvist.

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