Tidigare forskning har visat att elever har bristande kunskaper i multiplikationstabellerna. Att automatisera tabellerna ger eleverna goda förutsättningar inför övriga matematikområden. Syftet med studien var att skapa kunskap om lärares uppfattning till automatisering av multiplikationstabellerna samt att få kunskap om hur många av lärarnas elever som hade automatiserat tabellerna. Syftet var även att ta reda på vilka metoder lärare använder för att stötta eleverna i detta. Studien genomfördes med enkät som datainsamlingsinstrument och resultatet från enkäten följdes upp med fokusgruppsintervjuer. Resultatet av studien visade att många lärare upplever automatisering av multiplikationstabellerna som viktigt eftersom det underlättar för eleverna inför övriga matematikområden samt att det avlastar elevernas arbetsminne. Metoderna som flest lärare använde i sin undervisning var digitala undervisningsplattformar samt olika arbetsblad som drillträning av tabellerna. 

In this paper we derive and discuss some newtheorems related to all rearrangementsof a given set inRn, denoted(x)and use the results to prove some new Jensen type inequalitiesfor convex, superquadratic, strongly convex and 1-quasiconvex functions.

In this paper we discuss the Hermite-Hadamard and Fejer inequalities vis-a-vis the convexity concept. In particular, we derive some new theorems and examples where Hermite-Hadamard and Fejer type inequalities are satisfied without the assumptions of convexity or concavity on the actual interval [a,b]

In this paper, we develop explicit and semi-implicit second-order high-resolution finite difference schemes for a structured coagulation-fragmentation model formulated on the space of Radon measures. We prove the convergence of each of the two schemes to the unique weak solution of the model. We perform numerical simulations to demonstrate that the second order accuracy in the Bounded-Lipschitz norm is achieved by both schemes.

OBJECTIVE: To examine several dimensions of health-related quality of life (HRQL) in postmenopausal women who report previous fractures, and to provide perspective by comparing these findings with those in other chronic conditions (diabetes, arthritis, lung disease).

PATIENTS AND METHODS: Fractures are a major cause of morbidity among older women. Few studies have examined HRQL In women who have had prior fractures and the effect of prior fracture location on HRQL. In this observational study of 57,141 postmenopausal women aged 55 years and older (enrollment from December 2007 to March 2009) from 17 study sites in 10 countries, HRQL was measured using the European Quality of Life 5 Dimensions Index (EQ-5D) and the health status, physical function, and vitality questions of the Medical Outcomes Study 36-Item Short Form Health Survey (SF-36).

RESULTS: Reductions in EQ-5D health-utility scores and SF-36 measured health status, physical function, and vitality were seen in association with 9 of 10 fracture locations. Spine, hip, and upper leg fractures resulted in the greatest reductions In quality of life (EQ-5D scores, 0.62, 0.64, and 0.61, respectively, vs 0.79 without prior fracture). Women with fractures at any of these 3 locations, as well as women with a history of multiple fractures (EQ-5D scores, 0.74 for 1 prior fracture, 0.68 for 2, and 0.58 for >= 3), had reductions in HRQL that were similar to or worse than those in women with other chronic diseases (0.67 for diabetes, 0.69 for arthritis, and 0.71 for lung disease).

CONCLUSION: Previous fractures at a variety of bone locations, particularly spine, hip, and upper leg, or involving more than 1 location are associated with significant reductions in quality of life.

Studiens syfte var att ta reda på vilka strategier lärare som använder en Montessori-inspirerad pedagogik anser stimulerar och utmanar högpresterande elever i matematik samt om de upplevde några svårigheter eller problem med att undervisa högpresterande elever i matematik. För att undersöka det har sex kvalitativa intervjuer genomförts med lärare som har varit eller som är verksamma i årskurserna 1-3. Resultatet av intervjuerna visar att lärarna har en gemensam uppfattning om att en Montessori-inspirerad pedagogik utmanar och stimulerar högpresterande elever i matematik eftersom grundtanken i Montessori-pedagogiken är "eleven i centrum". Det här innebär att lärarna måste utgå från varje elevs förutsättningar, behov och aktuella kunskapsnivå. I resultatet framkommer också att lärarna använder sig av olika strategier i form av matematikbok från en högre årskurs, special- och fördjupningsområden, åldersintegrering och lärande med flera sinnen för att stimulera och utmana de högpresterande eleverna matematiskt. Det framgår också i resultatet att lärarna inte upplever några större problem eller svårigheter i undervisningen av högpresterande elever i matematik. De belyser ändå att den egna matematiska kompetensen, tidsbrist och eget kontrollbehov ibland kan utgöra hinder för att stimulera och utmana de högpresterande eleverna tillräckligt. Utifrån studien kan man dra slutsatserna att lärare som använder en Montessori-inspirerad pedagogik lägger stor vikt vid att hitta rätt strategi för rätt elev, vilket bidrar till att de högpresterande eleverna får utmaning och stimulans. De olika hindren som lärarna belyser skulle kunna minskas med fortbildning och samarbete mellan årskurserna. 

The starting point of this work has been to get knowledge about teachers working in a mathematical workshop with the students. I also wanted to investigate what students think about math, both within the workshop and in traditional teaching.

To achieve this I chose to interview a teacher in charge within the framework of a mathematical workshop, and six students in the 5^th grade. I also did a lesson observation with the purpose to see how well the teacher’s tutoring agrees with the results of the interview.

In this study I have discovered that the lessons in the mathematical workshop are planned after what the students are working with in the textbook. The teacher opens the lessons with the whole class and then splits it up in groups. In the interview the respondent conveys the importance of discussions among the students where they can explain their thoughts, especially for the weaker students that, thanks to the discussions are able to show their knowledge more. This is something I did not see during my lesson observation, however. The students also seemed to miss this when I spoke to them in the interview. They describe that they cannot see the connection between what they learn in the workshop and the textbook.

This work proposes the complete design cycle for several auxetic materials where the cycle consists of three steps (i) the design of the micro-architecture, (ii) the manufacturing of the material and (iii) the testing of the material. We use topology optimization via a level-set method and asymptotic homogenization to obtain periodic micro-architectured materials with a prescribed effective elasticity tensor and Poisson’s ratio. The space of admissible micro-architectural shapes that carries orthotropic material symmetry allows to attain shapes with an effective Poisson’s ratio below −1−1. Moreover, the specimens were manufactured using a commercial stereolithography Ember printer and are mechanically tested. The observed displacement and strain fields during tensile testing obtained by digital image correlation match the predictions from the finite element simulations and demonstrate the efficiency of the design cycle.

We design the micro-architecture of thin elastic panels to control their macroscopic behavior, accounting simultaneously for in-plane stiffness, out-of-plane stiffness and the extension–bending coupling effects. Our topology optimization method combines inverse homogenization, the Hadamard shape derivative and a level set method in the diffuse interface context to systematically capture within the unit cell the optimal micro-architecture. The efficiency of the solution method is illustrated through four numerical examples where the designed shape yields an important extension–bending coupling. The deformation responses under tensile loading are assessed numerically both on the complete periodic panel and on its homogenized twin plate. The results demonstrate that the simultaneous control of the in-plane, out-of-plane and their coupled behavior enables to shift a flat panel into a dome or a saddle shaped structure. Moreover, the obtained unit cells are elementary blocks to create directly 3D printable objects with shape-morphing capabilities.

Abstract

The purpose of this study is to investigate how teachers and study supervisors for year 1-3 teach newly arrived students’ in mathematical word problems and

their views on the students’ challenges. The method used for this study is

qualitative semi-structures interviews aimed at teachers and study supervisors. The study is based upon ten interviews, six teachers and four study supervisors. The sociocultural perspective on learning is used, as this perspective is implemented in Swedish school system.

The result of the study show the importance of the Swedish language for newly arrived students to be able to take on mathematical word problems. Furthermore, the result shows that the difficulties for newly arrived students lie mainly in their language skills rather than their mathematical skills. Previous research and the result of the interviews also confirm this and emphasize the importance of study guidance in the students’ native language.

Keywords

Word problems, socio-cultural perspective, newly arrived students, bilingual teaching

We consider the initial boundary value problem for the beam equation with the nonlinear strain. In our previous work this problem was proposed as a mathematical model for stretching and shrinking motions of the curve made of the elastic material on the plane. The aim of this paper is to establish uniqueness and existence of weak solutions. In particular, the uniqueness is proved by applying the approximate dual equation method.

In this article, we propose a macro-micro (two-scale) mathematical model for describing the macroscopic swelling of a rubber foam caused by the microscopic absorption of some liquid. In our modeling approach, we suppose that the material occupies a one-dimensional domain which swells as described by the standard beam equation including an additional term determined by the liquid pressure. As special feature of our model, the absorption takes place inside the rubber foam via a lower length scale, which is assumed to be inherently present in such a structured material. The liquid's absorption and transport inside the material is modeled by means of a nonlinear parabolic equation derived from Darcy's law posed in a non-cylindrical domain defined by the macroscopic deformation (which is a solution of the beam equation). Under suitable assumptions, we establish the existence and uniqueness of a suitable class of solutions to our evolution system coupling the nonlinear parabolic equation posed on the microscopic non-cylindrical domain with the beam equation posed on the macroscopic cylindrical domain. In order to guarantee the regularity of the non-cylindrical domain, we impose a singularity to the elastic response function appearing in the beam equation. 

Sato-Aiki-Murase-Shirakawa [7, 16] proposed a free boundary problem in order to describe water adsorption appearing in moisture transport of concrete carbonation process. The problem is to find a pair of a curve and a function for given boundary and initial functions. In this paper we consider that the boundary and initial functions depend on some parameter. Kumazaki [9, 10] already proved the continuous and measurable properties of solutions with respect to the parameter. The aim of this paper is to establish differentiability of solutions with respect to the parameter as a new property. This result will be applied to research of the multi-scale model to water adsorption in concrete carbonation.

In many industrial applications, rubber-based materials are routinely used in conjunction with various penetrants or diluents in gaseous or liquid form. It is of interest to estimate theoretically the penetration depth as well as the amount of diffusants stored inside the material. In this framework, we prove the global solvability and explore the large time-behavior of solutions to a one-phase free boundary problem with nonlinear kinetic condition that is able to describe the migration of diffusants into rubber. The key idea in the proof of the large time behavior is to benefit from a contradiction argument, since it is difficult to obtain uniform estimates for the growth rate of the free boundary due to the use of a Robin boundary condition posed at the fixed boundary.

We study the large-time behavior of the free boundary position capturing the one-dimensional motion of the carbonation reaction front in concrete-based materials. We extend here our rigorous justification of the t-behavior of reaction penetration depths by including nonlinear effects due to deviations from the classical Henry's law and time-dependent Dirichlet data.

We prove the large time behavior of solutions to a coupled thermo-diffusion arising in the modeling of the motion of hot colloidal particles in porous media. Additionally, we also ensure the uniqueness of solutions of the target problem. The main mathematical difficulty is due to the presence in the right-hand side of the equations of products between temperature and concentration gradients. Such terms mimic the so-called thermodynamic Soret and Dufour effects. These are cross-coupling terms emphasizing in this context a strong interplay between heat conduction and molecular diffusion.

In our previous works we studied a one-dimensional free-boundary model related to the aggressive penetration of gaseous carbon dioxide in unsaturated concrete. Essentially, global existence and uniqueness of weak solutions to the model were obtained when the initial functions are bounded on the domain. In this paper we investigate the well-posedness of the problem for the case when the initial functions belong to a  class. Specifically, the uniqueness of weak solutions is proved by applying the dual equation method.

The outbreak of COVID-19 disease caused by the SARS-CoV-2 in 2019 has claimed over 6.3 million lives. The pandemic prompted many countries to set up some preventive measures as means to control the spread of the disease. In this thesis, a deterministic system of coupled differential equations is proposed to study the transmission of COVID-19 among a well-mixed population with intervention strategies. The existence and uniqueness of the classical solution of the COVID-19 model are proved. The equilibrium points of the model are analyzed and, the basic reproduction number is obtained. The local asymptotic stability and the global asymptotic stability of the model are carried out. An adaptive Dormand\textendash Prince numerical method is used to obtain approximate solution of the model. The results shows that combined control parameters may reduce the burden of COVID-19 faster in the population. In addition, the outcomes of this study show that in order to mitigate the spread of COVID-19 in the overall population, non-pharmaceutical intervention strategies such as social distancing, self-isolation, and hand washing should be practiced at the maximum and people should be vaccinated.

In this paper we prove some essential complements of the paper (J. Inequal. Appl.2019:171, 2019) on the same theme. We prove some new Fourier inequalities in thecase of the Lorentz–Zygmund function spaces Lq,r(log L)α involved and in the casewith an unbounded orthonormal system. More exactly, in this paper we prove anddiscuss some new Fourier inequalities of this type for the limit case L2,r(log L)α, whichcould not be proved with the techniques used in the paper (J. Inequal. Appl.2019:171, 2019).

This paper is an essential complement of the research recently presented in [1], [2]. A number of classical Fourier inequalities related to Fourier coefficients with respect to unbounded orthogonal systems are generalized and complemented. All results are given in the case of generalized Lorentz spaces.

We consider the generalized Lorentz space L ;q dened via a continuous and concave function and the Fourier series of a function with respect to an unbounded orthonormal system. Some new Fourier and Jackson-Nikol'skii type inequalities in this frame are stated, proved and discussed. In particular, the derived results generalize and unify several well-known results but also some new applications are pointed out.

The Cauchy-Schwarz (C-S) inequality is one of the most famous inequalities in mathematics. In this survey article, we first give a brief history of the inequality. Afterward, we present the C-S inequality for inner product spaces. Focusing on operator inequalities, we then review some significant recent developments of the C-S inequality and its reverses for Hilbert space operators and elements of Hilbert C*-modules. In particular, we pay special attention to an operator Wielandt inequality.

Dagens snabba digitala utveckling påverkar samhället och inte minst skolan. Under de senare åren har skolan digitaliserats vilket bland annat har inneburit att digitala läromedel har utvecklats och börjar användas i undervisningen för att främja elevers lärande. Vår studie syftar till att undersöka i vilken utsträckning digitala läromedel som används i matematikundervisningen i årskurs 3 innehåller aktiviteter som möjliggör olika typer av samarbete för elever. För att besvara studiens syfte gjordes en innehållsanalys av tre utvalda digitala läromedel med tillhörande lärarhandledningar. Studiens resultat visar att de granskade digitala läromedlen med tillhörande lärarhandledningar ger få möjligheter till samarbete mellan elever. Analysen visar att de flesta samarbetsuppgifterna återfinns i lärarhandledningarna vilket innebär att läraren har en central roll för att möjliggöra tillfällen för samarbete. Analysen visar även att de samarbetsuppgifter som eleverna får arbeta med inte kräver någon gemensam gruppinsats. 

This thesis is devoted to the study of mixed norm spaces that arise in connection with embeddings of Sobolev and Besov type spaces. We study different structural, integrability, and smoothness properties of functions satisfying certain mixed norm conditions. Conditions of this type are determined by the behaviour of linear sections of functions. The work in this direction originates in a paper due to Gagliardo (1958), and was further developed by Fournier (1988), by Blei and Fournier (1989), and by Kolyada (2005).

Here we continue these studies. We obtain some refinements of known embeddings for certain mixed norm spaces introduced by Gagliardo, and we study general properties of these spaces. In connection with these results, we consider a scale of intermediate mixed norm spaces, and prove intrinsic embeddings in this scale.

We also consider more general, fully anisotropic, mixed norm spaces. Our main theorem states an embedding of these spaces to Lorentz spaces. Applying this result, we obtain sharp embedding theorems for anisotropic Sobolev-Besov spaces, and anisotropic fractional Sobolev spaces. The methods used are based on non-increasing rearrangements, and on estimates of sections of functions and sections of sets. We also study limiting relations between embeddings of spaces of different type. More exactly, mixed norm estimates enable us to get embedding constants with sharp asymptotic behaviour. This gives an extension of the results obtained for isotropic Besov spaces by Bourgain, Brezis, and Mironescu, and for anisotropic Besov spaces by Kolyada.

We study also some basic properties (in particular the approximation properties) of special weak type spaces that play an important role in the construction of mixed norm spaces, and in the description of Sobolev type embeddings.

In the last chapter, we study mixed norm spaces consisting of functions that have smooth sections. We prove embeddings of these spaces to Lorentz spaces. From this result, known properties of Sobolev-Liouville spaces follow.

This thesis is devoted to the study of mixed norm spaces that arise in connection with embeddings of Sobolev and Besov type spaces. The work in this direction originates in a paper due to Gagliardo (1958), and was continued by Fournier (1988) and by Kolyada (2005).

We consider fully anisotropic mixed norm spaces. Our main theorem states an embedding of these spaces into Lorentz spaces. Applying this result, we obtain sharp embedding theorems for anisotropic fractional Sobolev spaces and anisotropic Sobolev-Besov spaces. The methods used are based on non-increasing rearrangements and on estimates of sections of functions and sections of sets. We also study limiting relations between embeddings of spaces of different type. More exactly, mixed norm estimates enable us to get embedding constants with sharp asymptotic behaviour. This gives an extension of the results obtained for isotropic Besov spaces $B_p^\alpha$ by Bourgain, Brezis, and Mironescu, and for Besov spaces $B^{\alpha_1,\dots,\alpha_n}_p$ by Kolyada.

We study also some basic properties (in particular the approximation properties) of special weak type spaces that play an important role in the construction of mixed norm spaces and in the description of Sobolev type embeddings.

Syftet med studien var att analysera tid och klocka och dess progression i läromedel, för åk 4–6. Studien syftade även utifrån uppgifter kopplade till tid och klocka att analysera vilka repre-sentationsformer och vilka förmågor det är som tränas upp i de utvalda läromedlen. För att studera såväl som analysera olika matematikläromedel, användes ett dokument- och läromedelsa-nalys för att bygga upp tillvägagångssättet för studien.

Resultatet av studien visade på att de valda läromedlen hade en blandning av representationsformerna dock saknades ofta representationsformen konkret modell. Konkreta modeller fanns i några enstaka uppgifter. Studiens resultat visar även att eleven får möjlighet att träna de fem matematiska förmågor men i olika utsträckning. Kommunikations- och resonemangsförmågan är de förmågor som tränas minst i läromedel.

Slutsatsen som kan dras är att lärarens roll vid användning av läromedel i matematik är av stor vikt eftersom vissa förmågor synliggörs mer i arbetsboken än andra. När olika representat-ionsformer används hjälper det eleven att få grundläggande kunskap om tid och klockan. En annan slutsats är att resonemangsförmågan och kommunikationsförmågan behöver lyftas upp mer, så att eleven får möjlighet att resonera och samtala med varandra.

We consider a dynamical systems formulation for models with an exponential scalar field and matter with a linear equation of state in a spatially flat and isotropic spacetime. In contrast to earlier work, which only considered linear hyperbolic fixed point analysis, we do a center manifold analysis of the non-hyperbolic fixed points associated with bifurcations. More importantly though, we construct monotonic functions and a Dulac function. Together with the complete local fixed point analysis this leads to proofs that describe the entire global dynamics of these models, thereby complementing previous local results in the literature. 

We study flat Friedmann-Lemaitre-Robertson-Walker alpha-attractor E- and T-models by introducing a dynamical systems framework that yields regularized unconstrained field equations on two-dimensional compact state spaces. This results in both illustrative figures and a complete description of the entire solution spaces of these models, including asymptotics. In particular, it is shown that observational viability, which requires a sufficient number of e-folds, is associated with a particular solution given by a one-dimensional center manifold of a past asymptotic de Sitter state, where the center manifold structure also explains why nearby solutions are attracted to this "inflationary attractor solution." A center manifold expansion yields a description of the inflationary regime with arbitrary analytic accuracy, where the slow-roll approximation asymptotically describes the tangency condition of the center manifold at the asymptotic de Sitter state.

It is impossible in limited number of pages to give a fair picture of such a remarkable man, great mathematician and human being as Josip Pecaric. Our intention is instead to complement the picture of him in various ways. We hope that our paper will give also someflavor of Josip as family man, fighter, supervisor,international authority, author (also in other subjects than mathematics), fan of the Croatian football team, and not only as his obvious role as our King of Inequalities.

In this paper, we derive sharp lower bounds, also known as quantum speed limits, for the time it takes to transform a quantum system into a state such that an observable assumes its lowest average value. We assume that the system is initially in an incoherent state relative to the observable and that the state evolves according to a von Neumann equation with a Hamiltonian whose bandwidth is uniformly bounded. The transformation time depends intricately on the observable's and the initial state's eigenvalue spectrum and the relative constellation of the associated eigenspaces. The problem of finding quantum speed limits consequently divides into different cases requiring different strategies. We derive quantum speed limits in a large number of cases, and we simultaneously develop a method to break down complex cases into manageable ones. The derivations involve both combinatorial and differential geometric techniques. We also study multipartite systems and show that allowing correlations between the parts can speed up the transformation time. In a final section, we use the quantum speed limits to obtain upper bounds on the power with which energy can be extracted from quantum batteries.

We consider multirate iterative schemes for the Biot system modeling coupled flow and geomechanics in a poro-elastic medium. The multirate iterative coupling scheme exploits the different time scales for the mechanics and flow problems by taking multiple finer time steps for flow within one coarse mechanics time step. We adapt the fixed stress split algorithm that decouples the flow and mechanics equations for the multirate case and perform an iteration between the two problems until convergence. We provide a fully discrete scheme that uses Backward Euler time discretization and mixed spaces for flow and conformal Galerkin for mechanics. Our analysis is based on studying the equations satisfied by the difference of iterates and using Banach contraction argument to prove that the corresponding scheme is a fixed point contraction. The analysis provides the value of an adjustable coefficient used in the proposed iterative coupling algorithms. Furthermore, we show that the converged quantities satisfy the variational weak form for the coupled discrete system. (C) 2016 Elsevier B.V. All rights reserved.

We consider single rate and multirate explicit schemes for the Biot system modeling coupled flow and geomechanics in a poro-elastic medium. These schemes are widely used in practice that follows a sequential procedure in which the flow and mechanics problems are fully decoupled. In such a scheme, the flow problem is solved first with time-lagging the displacement term followed by the mechanics solve. The multirate explicit coupling scheme exploits the different time scales for the mechanics and flow problems by taking multiple finer time steps for flow within one coarse mechanics time step. We provide fully discrete schemes for both the single and multirate approaches that use Backward Euler time discretization and mixed spaces for flow and conformal Galerkin for mechanics. We perform a rigorous stability analysis and derive the conditions on reservoir parameters and the number of finer flow solves to ensure stability for both schemes. Furthermore, we investigate the computational time savings for explicit coupling schemes against iterative coupling schemes.

In this paper, we consider an iterative coupling scheme for solving a fully discretized Biot system based on the fixed-stress split coupling algorithm. Specifically, we derive a priori error estimates for quantifying the error between the solution obtained at any iterate and the true solution. Our approach is based on studying the equations satisfied by the difference of iterates and utilizing a Banach contraction argument to show that the corresponding scheme is a fixed point iteration. Obtained contraction results are then used to derive theoretical convergence error estimates for the single rate iterative coupling scheme. We compare our numerical computations against the theoretically derived contraction estimates and show a good agreement with theory.

Recently, an accurate coupling between subsurface flow and reservoir geomechanics has received more attention in both academia and industry. This stems from the fact that incorporating a geomechanics model into upstream flow simulation is critical for accurately predicting wellbore instabilities and hydraulic fracturing processes. One of the recently introduced iterative coupling algorithms to couple flow with geomechanics is the undrained split iterative coupling algorithm as reported by Kumar et al. (2016) and Mikelic and Wheeler (Comput. Geosci. 17: 455–461 2013). The convergence of this scheme is established in Mikelic and Wheeler (Comput. Geosci. 17:455–461 2013) for the single rate iterative coupling algorithm and in Kumar et al. (2016) for the multirate iterative coupling algorithm, in which the flow takes multiple finer time steps within one coarse mechanics time step. All previously established results study the convergence of the scheme in homogeneous poroelastic media. In this work, following the approach in Almani et al. (2017), we extend these results to the case of heterogeneous poroelastic media, in which each grid cell is associated with its own set of flow and mechanics parameters for both the single rate and multirate schemes. Second, following the approach in Almani et al. (Comput. Geosci. 21:1157–1172 2017), we establish a priori error estimates for the single rate case of the scheme in homogeneous poroelastic media. To the best of our knowledge, this is the first rigorous and complete mathematical analysis of the undrained split iterative coupling scheme in heterogeneous poroelastic media.