The main result of this paper is that for the har dsphere kernel, the solution of the spatially homogenous Boltzmann equation converges strongly in L1 to equilibrium given that the initial data f0 belongs to L1(R3,(1+v^2)dv). This was previously known to be true with the additional assumption that f0logf0 belonged to L1(R3), which corresponds to bounded initial entropy.

In the so called outer polygon method, an approximative conformal mapping for a given simply connected region Ω is constructed using a Schwarz–Christoffel mapping for an outer polygon, a polygonal region of which Ω is a subset. The resulting region is then bounded by a C^∞-curve, which among other things means that its curvature is bounded.In this work, we study the curvature of an inner curve in a polygon, i.e., the image under the Schwarz–Christoffel mapping from R, the unit disk or upper half–plane, to a polygonal region P of a curve inside R. From the Schwarz–Christoffel formula, explicit expressions for the curvature are derived, and for boundary curves, appearing in the outer polygon method, estimations of boundaries for the curvature are given.

We develop a cut finite element method for the Bernoulli free boundary problem. The free boundary, represented by an approximate signed distance function on a fixed background mesh, is allowed to intersect elements in an arbitrary fashion. This leads to so called cut elements in the vicinity of the boundary. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements penalizing the gradient jumps across element sides. The stabilization also ensures good conditioning of the resulting discrete system. We develop a method for shape optimization based on moving the distance function along a velocity field which is computed as the H¹ Riesz representation of the shape derivative. We show that the velocity field is the solution to an interface problem and we prove an a priori error estimate of optimal order, given the limited regularity of the velocity field across the interface, for the velocity field in the H¹norm. Finally, we present illustrating numerical results.

We design and analyse a Nitsche method for contact problems. Compared to the seminal work of Chouly and Hild [SIAM J. Numer. Anal., 51 (2013), pp. 1295--1307], our method is constructed by expressing the contact conditions in a nonlinear function for the displacement variable instead of the lateral forces. The contact condition is then imposed using the nonsymmetric variant of Nitsche's method that does not require a penalty term for stability. Nonconforming piecewise affine elements are considered for the bulk discretization. We prove optimal error estimates in the energy norm.

We develop a discontinuous cut finite element method for the Laplace–Beltrami operator on a hypersurface embedded in R. The method is constructed by using a discontinuous piecewise linear finite element space defined on a background mesh in R. The surface is approximated by a continuous piecewise linear surface that cuts through the background mesh in an arbitrary fashion. Then, a discontinuous Galerkin method is formulated on the discrete surface and in order to obtain coercivity, certain stabilization terms are added on the faces between neighbouring elements that provide control of the discontinuity as well as the jump in the gradient. We derive optimal a priori error and condition number estimates which are independent of the positioning of the surface in the background mesh. Finally, we present numerical examples confirming our theoretical results.

Most pupils today lack understanding of how to add fractions of how to add fractions using various methods, despite several years of instruction in fraction arithmetic. The study aims to generate a deeper understanding of the previously identified critical aspects encountered by fourth-grade pupils in addition calculations involving fractions. The study has two main research questions: 1) How are the preciously identified critical aspects manifested in fourth-grade pupils’ solutions and reasoning regarding addition calculations with fraction? 2) Which critical aspects emerge in pupils’ solutions using numerical and visual representations? 

The study employs a combination of quantitative and qualitative methods, integrating elements of variation theory. The applied methods include the use of worksheet and conducting semi-structured interviews. A total of 35 pupils participated in the investigation, with 6 pupils selected for interviews. The results of the study identified three critical aspects. Critical aspect 1: Discerning how addition calculations with fractions differ from those involving whole numbers. Critical aspect 2:  Discerning how addition calculations with fractions are preformed when the denominators are different. Critical aspect 3: Discerning how addition calculations are carried out using representations of fractions as parts of a whole. These critical aspects, which have been highlighted in previous studies, confirm, and support the results of the study. 

What is the most important aspect when shelves are being sold, manufactured and used? Well they must be user-friendly which means they should be easy to assemble, carry and modify. They should also be easy to ship, weight aspect and not ungainly. Most importantly it must be worth its price or just cheap, to accomplish this one should keep the amount of material to a minimum whilst the manufacturing must be easy.

These were some of the questions at issue that this work has started from and they have more or less thoroughly been penetrated.

In an effort to acquire some answers the possibility to reduce the thickness of the poles to the shelves from 2,5 mm to 2,0 mm, this would present partial solutions to both weight and material issues but before all the price issue though the material contributes to a large piece of the cost. This suggestion appeared when the poles were investigated and calculated from different angels to determine if they would hold out to this kind of change. The angels that were investigated are bending of the pole, the holes in the pole will be sheared (ripped like can opener-effect) by the hooks on the shelf consol and the detachable foot. The result that emerged are that some sizes of the poles can endure the change without any thought whilst the smaller poles requires some further testing and pondering before implementing. The aspect that was found to be norm setting was the bending of the pole.

The fact that the user friendliness always can be improved a helper was designed. The helper is meant for the user is so he/she easily can see how much merchandise (weight) a shelf and pole can withstand. In other words to make sure that the shelf/pole is not overloaded and risk a breakdown.

To declare where Itabs system is in regards to its competitor on these important questions a competitor analysis was conducted. The primary test-issues were stability, easy to assemble and if there are any apparent hazardous areas on the stand. Unfortunately there were some difficulties in retrieving the stands from some of the competitors or more importantly the stand from Itab themselves. So a comparison was not possible although they who were interested in testing their stands and had sent them in were to be pleased so a test was conducted. The issue regarding the price was not compared though this report is not meant for the users in the stores.

We consider a stabilized finite element method for the Darcy problem on a surface based on the Masud–Hughes formulation. A special feature of the method is that the tangential condition of the velocity field is weakly enforced through the bilinear form, and that standard parametric continuous polynomial spaces on triangulations can be used. We prove optimal order a priori estimates that take the approximation of the geometry and the solution into account.

This thesis project has been written and accomplished in cooperation with HABO AB. The objective of the project was to develop a standard stated packaging system. The background to this is that HABO is not pleased with the system currently in use. The system which HABO is currently using consists of to many sizes. The largest benefit of a standard stated system is that it becomes much easier to organize the packaging with the aid of a computer program. If one manages to standardize the packaging system if will be much easier to calculate how much space that is required for the products in the store. It also enables HABO to sell pre defined product groups.

The basic condition for the system is that it should be compatible with Hestra wall panels and store equipment. The store equipment proved to be very determining during the project. Due to its design the number of possible variations of sizes decreased considerably.

Much of the work has been simulated in a computer model which has been constructed in ProEngineer. With aid of the model suitable packaging sizes could be tested on Hestras wall panels.

The work resulted in a system with three different width sizes (74, 116 and 158 mm), compared to the present five. The number of lengths was set to four (118, 182, 246 and 374 mm) in comparison with today where the length is decided for each product individually. In order for the system to work properly it is crucial the HABOs suppliers can guarantee a specified length for all of the different packaging sizes.

The basic principle is that both SB-pack and Skin-pack should be possible to combine to achieve the most effective system.

A couple of improvements were also considered. Standard setting of the width proved to be the hardest challenge. Therefore an alternative could be to use a gliding spear instead of Hestras traditional panel. This allows the spear to be continuously variable. This allows more freedom when it comes to designing the packaging. Regarding different types of packaging SB-pack with long back or colored plastic bag is preferred instead of the traditional SB-pack. The products are displayed in a better fashion and it gives a better impression.

In grades 4–6, students must undergo a progression to learn how rational numbers and numbers in decimal form are structured. Research has shown that there can be several difficulties and misconceptions about the area, including knowing the different place value of numbers. This study has been inspired by a previous study, conducted by Jarl and Johansson (2014). The aim of this study is to compare whether the same critical aspects identified in Jarl and Johansson (2014) studies also show up in other student groups. The subject of interest in this study was: What critical aspects can be identified in a grade 4 and a grade 5 around numbers in decimal form?

In order to be able to answer the question, the students in this study have had to complete a worksheet with tasks linked to numbers in decimal form. Thereafter, qualitative interviews were conducted to gain a broader insight into what critical aspects the students have or have not distinguished. The choice of method in the study has elements of the theory of variation where the students need to see the necessary details (in the study called critical aspects). The results of the study show that all critical aspects that were identified in Jarl and Johansson's (2014) study were also critical in this study. However, a new critical aspect was identified: Students need to understand that numbers on each side of the decimal point together must become a number. Knowledge of critical aspects can be seen as special knowledge for teachers to know what can be misunderstood about the mathematical field. This knowledge cannot generalize, but it can be transferable so that a critical aspect can be identified in other student groups.

In stochastic frontier analysis, the conventional estimation of unit inefficiency is based on the mean/mode of the inefficiency, conditioned on the composite error. It is known that the conditional mean of inefficiency shrinks towards the mean rather than towards the unit inefficiency. In this paper, we analytically prove that the conditional mode cannot accurately estimate unit inefficiency, either. We propose regularized estimators of unit inefficiency that restrict the unit inefficiency estimators to satisfy some a priori assumptions, and derive the closed form regularized conditional mode estimators for the three most commonly used inefficiency densities. Extensive simulations show that, under common empirical situations, e.g., regarding sample size and signal-to-noise ratio, the regularized estimators outperform the conventional (unregularized) estimators when the inefficiency is greater than its mean/mode. Based on real data from the electricity distribution sector in Sweden, we demonstrate that the conventional conditional estimators and our regularized conditional estimators provide substantially different results for highly inefficient companies.