Detta arbete har undersökt hur många betjäningsstationer avdelningen Byten och Återköp på IKEA Kungens Kurva behöver för att nå en förväntad kötid inom nio minuter.  Arbetet är uppdelat i en kvantitativ del samt en kvalitativ del. Den kvantitativa delen av arbetet besvarade den matematiska frågeställningen där kön var modellerad enligt ett M|M|c-­‐system med Poissonfördelade ankomstintensiteter samt exponentialfördelade betjäningsintensiteter.  Resultatet var baserat på data från januari 2015.  I första hand reglerades antal betjäningsstationer för att nå en förväntad kötid under nio minuter.  I andra hand reglerades betjäningsintensiteten, om kötiden ännu inte uppnåtts.  Den kvalitativa delen av arbetet baserades på ett teoretiskt ramverk gällande Customer Relationship Management samt intervjuer med Avdelningschefen och Kundrelationschefen på IKEA Kungens Kurva.  Arbetets slutsats baserades på resultat från både den kvantitativa och den kvalitativa delen av arbetet. Det matematiska resultatet presenterar antalet öppna betjäningsstationer som krävdes timme för timme under vardagar respektive helger för att nå en förväntad kötid under nio minuter.  Slutsatsen är att i de fall där den förväntade kötiden var strax över nio minuter kommer detta inte påverka kundrelationen, så länge en god betjäning ges, medan det matematiska resultatet bör tillämpas om den förväntade kötiden är långt över nio minuter. / The main aim of conducting this bachelor thesis has been to examine how many service stations the Exchange and Returns Department at IKEA Kungens Kurva need in order to have an expected queuing time below nine minutes.  The thesis was divided in a quantitative and a qualitative study.  The quantitative study solved the mathematical research question where the queue was modelled as a M|M|c queue with Poisson distributed arrival intensities and exponential distributed service intensities.  This was based on historical data from January 2015 for previous queuing times at the department.  Firstly the number of service stations was adjusted in order to reach the given expected queuing time.  Secondly, if the goal still was not reached the service intensity was adjusted.  The qualitative study was based on a theoretical framework on Customer Relationship Management and interviews with the Department Manager as well as the Customer Relations Manager at IKEA Kungens Kurva.  The conclusion of the bachelor thesis was constructed by the results from both the quantitative and the qualitative studies.  The mathematical result presents the number of service stations needed every hour for weekdays and weekends, in order to have an expected queuing time below nine minutes.  The conclusion of the thesis stated that if the queuing time was just above nine minutes, it would not affect the customer relations as long as good service was given.  In order to increase the customer value at the department, IKEA Kungens Kurva needs to center the attention on the service quality, but also adjust the number of service stations according to the mathematical result when the expected queuing time was far over nine minutes.

This Licentiate thesis deals with the study of existence and uniqueness together with coercive estimates for solutions of certain differential equations. The thesis consists of four papers (papers A, B, C and D) and an introduction, which put these papers into a more general frame and which also serves as an overview of this interesting field of mathematics. In the text below the functions r(x), q(x), m(x) etc. are functions on (-∞,+∞), which are different but well defined in each paper. In paper A we study the separation and approximation properties for the differential operator ly=-y″+r(x)y′+q(x)y in the Hilbert space L2 :=L2(R), R=(-∞,+∞), as well as the existence problem for a second order nonlinear differential equation in L2 . Paper B deals with the study of separation and approximation properties for the differential operator ly=-y″+r(x)y′+s(x)‾y′ in the Hilbert spaceL2:=L2(R), R=(-∞,+∞), (here ¯y is the complex conjugate of y). A coercive estimate for the solution of the second order differential equation ly =f is obtained and its applications to spectral problems for the corresponding differential operatorlis demonstrated. Some sufficient conditions for the existence of the solutions of a class of nonlinear second order differential equations on the real axis are obtained. In paper C we study questions of the existence and uniqueness of solutions of the third order differential equation (L+λE)y:=-m(x)(m(x)y′)″+[q(x)+ir(x)+λ]y=f(x), (0.1) and conditions, which provide the following estimate: ||m(x)(m(x)y′)″||pp+||(q(x)+ir(x)+λ)y||pp≤ c||f(x)||pp for a solution y of (0.1). Paper D is devoted to the study of the existence and uniqueness for the solutions of the following more general third order differential equation with unbounded coefficients: -μ1(x)(μ2(x)(μ1(x)y′)′)′+(q(x)+ir(x)+λ)y=f(x). Some new existence and uniqueness results are proved and some normestimates of the solutions are given.

This thesis explores the application of Graph Neural Networks (GNNs) for forecasting net-positions in the Nordic electricity market. Two GNN architectures, GRU-GCN and FGNN, were evaluated and compared to the existing forecasting model employed in the power grid. Results demonstrate that both GNN models achieve competitive performance, highlighting their potential for leveraging the graph structure inherent in power grids. However, regional variations in forecast uncertainty and the impact of data quality and disruptions necessitate further research. This thesis contributes to the understanding of GNNs in power grid forecasting and identifies future research directions, such as developing interpretable GNN models and incorporating additional data sources, to enhance the accuracy and reliability of power grid operations.

QC 20120320

<p>This thesis consists of four papers about the Minkowski measure of asymmetry and the Minkowski (or Banach-Mazur) distance for convex bodies.We relate these two quantities by giving estimates for the Minkowski distance in terms of the Minkowski measure. We also investigate some properties of the Minkowski measure, in particular a stability estimate is given. More specifically, let <i>C</i> and <i>D</i> be n-dimensional convex bodies. Denote by As(<i>C</i>) and As(<i>D</i>) the Minkowski measures of asymmetry of <i>C</i> and <i>D </i>resp. and by <i>d</i>(<i>C,D</i>) the Minkowski distance between <i>C</i> and <i>D</i>.</p><p>In Paper I, by using a linearisation method for affine spaces and affine maps and using a generalisation of a lemma of D.R. Lewis, we proved that <i>d</i>(<i>C</i>,<i>D</i>) < <i>n</i>(As(<i>C</i>) + As(<i>D</i>))/2 for all convex bodies <i>C,D</i>.</p><p>In Paper II, by first proving some general existence theorems for a class of volume-increasing affine maps, we obtain the estimate that under the same conditions as in paper I, <i>d</i>(<i>C,D</i>) < (<i>n</i>-1) min(As(<i>C</i>),As(<i>D</i>)) + <i>n</i>.</p><p>In Paper III we consider the Minkowski measure itself. We determine the Minkowski measures for convex hulls of sets of the form <i>conv</i>(<i>C,p</i>) where <i>C</i> is a convex set with known measure of asymmetry and <i>p</i> is a point outside <i>C</i>.</p><p>In Paper IV, we focus on estimating the deviation of a convex body C from the simplex S if the Minkowski measure of C is close to the maximum value n (known to be attained only for the simplex). We prove that if As(C) > n - ε for 0 < ε < 1/δ where δ = 8(n+1), then d(C,S) < 1 + 8(n+1) ε .</p>

<p>This thesis consists of three papers on topics related to optimization and estimation of solutions of Riccati equations. We are concerned with the initial value problem</p><p><i>f</i>'+<i>f</i>² =<i>r</i>², <i>f</i>(0)=0, (*)</p><p>and we want to optimise</p><p><i>F</i>(<i>T</i>)= ∫₀^T <i>f</i>(<i>t</i>) <i>dt</i></p><p>when <i>r</i> is allowed to vary over the set <i>R</i>(φ ) of all <i>equimeasurable</i> rearrangements of a decreasing function φ and its convex hull <i>CR</i>(φ). </p><p>In the second paper we give a new proof of a lemma of Essén giving lower and upper bounds for the solution to the above equation, when <i>r</i> is increasing. We also generalize the lemma to a more general equation.</p><p>It was proved by Essén that the infimum of <i>F</i>(<i>T</i>) over <i>R</i>(φ) and <i>RC</i>(φ) is attained by the solution <i>f</i> of (*) associated to the increasing rearrangement of an element in <i>R</i>(φ). The supremum of <i>F</i>(<i>T</i>) over <i>RC</i>(φ) is obtained for the solution associated to a decreasing function <i>p</i>, though not necessarily the decreasing rearrangement φ, of an element in <i>R</i>(φ). By changing the perspective we determine the function <i>p </i>that solves the supremum problem.</p>

This thesis consists of four papers about the Minkowski measure of asymmetry and the Minkowski (or Banach-Mazur) distance for convex bodies.We relate these two quantities by giving estimates for the Minkowski distance in terms of the Minkowski measure. We also investigate some properties of the Minkowski measure, in particular a stability estimate is given. More specifically, let C and D be n-dimensional convex bodies. Denote by As(C) and As(D) the Minkowski measures of asymmetry of C and D resp. and by d(C,D) the Minkowski distance between C and D. In Paper I, by using a linearisation method for affine spaces and affine maps and using a generalisation of a lemma of D.R. Lewis, we proved that d(C,D) &lt; n(As(C) + As(D))/2 for all convex bodies C,D. In Paper II, by first proving some general existence theorems for a class of volume-increasing affine maps, we obtain the estimate that under the same conditions as in paper I, d(C,D) &lt; (n-1) min(As(C),As(D)) + n. In Paper III we consider the Minkowski measure itself. We determine the Minkowski measures for convex hulls of sets of the form conv(C,p) where C is a convex set with known measure of asymmetry and p is a point outside C. In Paper IV, we focus on estimating the deviation of a convex body C from the simplex S if the Minkowski measure of C is close to the maximum value n (known to be attained only for the simplex). We prove that if As(C) &gt; n - ε for 0 &lt; ε &lt; 1/δ where δ = 8(n+1), then d(C,S) &lt; 1 + 8(n+1) ε .

This thesis consists of three papers on topics related to optimization and estimation of solutions of Riccati equations. We are concerned with the initial value problem f'+f² =r², f(0)=0, (*) and we want to optimise F(T)= ∫0T f(t) dt when r is allowed to vary over the set R(φ ) of all equimeasurable rearrangements of a decreasing function φ and its convex hull CR(φ). In the second paper we give a new proof of a lemma of Essén giving lower and upper bounds for the solution to the above equation, when r is increasing. We also generalize the lemma to a more general equation. It was proved by Essén that the infimum of F(T) over R(φ) and RC(φ) is attained by the solution f of (*) associated to the increasing rearrangement of an element in R(φ). The supremum of F(T) over RC(φ) is obtained for the solution associated to a decreasing function p, though not necessarily the decreasing rearrangement φ, of an element in R(φ). By changing the perspective we determine the function p that solves the supremum problem.

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This essay is a detailed survey of an article from 1996 published by Franc Forstneric, where he studies symplectic automorphisms of C2n. The vision is to introduce the density property for holomorphic symplectic manifolds. Our idea is that of Dror Varolin when he in 2001 introduced the concept of density property for Stein manifolds. The main result here is the introduction of symplectic shears on C2n equipped with a holomorphic symplectic form and to show that the group generated by finite compositions of symplectic shears is dense in the group of symplectic automorphisms of C2n in the compact-open topology. We give a complete background of the tools from the theory of ordinary differential equations, smooth manifolds, and complex and symplectic geometry that is needed in order to prove this result. / Den här uppsatsen är en detaljerad undersökning av en artikel från 1996 publicerad av Franc Forstneric där han studerar symplektiska automorfismer av C2n. Visionen är att introducera täthetsegenskapen för holomorfa symplektiska mångfalder. Våran idé är som den av Dror Varolin när han 2001 introducerade täthetsegenskapen för Stein mångfalder. Huvudresultatet här är införandet av symplektiska skjuvningar på C2n med en holomorfisk symplektisk form och att visa att gruppen som genereras av ändliga sammansättningar av symplektiska skjuvningar är tät i gruppen av symplektiska automorfismer av C2n i den kompakt-öppna topologin. Vi ger en fullständig bakgrund av de verktyg från teorin om ordinära differentialekvationer, släta mångfalder och komplex och symplektisk geometri som behövs för att visa detta.