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This Thesis is based on five papers, four of which are published and one has been submitted for publication. For convenience of the reader, we also include one chapter that contains a brief overview of the results, some relevant background material, and comments on methods used in the proofs. The Thesis studies several problems from modern Harmonic analysis. More precisely, we focus on studying various properties of Paley-Wiener spaces: sampling sets, extreme and exposed points of the unit ball, and contraction operators.

Bridgeland stability conditions are powerful tools for studying derived categories, with several applications to algebraic geometry. They were introduced by Bridgeland in 2002 [Bri07], who was motivated by Douglas’ work on Π-stability of D-branes [Dou02] in the context of string theory. Bridgeland showed that the set Stab(D) of stability conditions on a triangulated category D is a complex manifold, a result of extreme importance and central to all mathematical applications of this field of study. But in order to use this concept of stability conditions in string theory (as intended by Bridgeland), one needs to prove the existence of stability conditions on the bounded derived category Db(X) of a compact Calabi-Yau threefold X. This task is far from easy, as it took more than a decade before the first example was produced for the smooth quintic threefold by Li in [Li18]. This achievement came into fruition thanks to the extensive amount of work in the domain over this period of time, where the existence of stability conditions was progressively established for arbitrary smooth projective varieties of dimension one [Bri07, Oka06, Mac07], dimension two [Bri08, AB13], and then some dimension three cases (see Section 1.3). One of the main applications of stability conditions on Db(X) (for an arbitrary variety X) is to study the geometry of moduli spaces of coherent sheaves over X with some Chern character v via the strategy known as “wall crossing”. In loose terms, a “wall” is a codimension one submanifold of Stab(Db(X)) such that by changing stability conditions along a continuous path in Stab(Db(X)) that goes through the wall causes the moduli space of sheaves over X with Chern character v to transform. When X is of dimension two, we have a solid control over wall crossing thanks to Bayer–Macr`ı [BM11], who provided a full understanding of how moduli spaces of sheaves change as we cross walls, as well as knowing the exact geometrical relationship these walls have with the underlying surface. In addition the precise structure of the walls is known and there are effective techniques to detect them. This thorough picture of wall crossing in dimension two is demonstrated through various complete studies of moduli spaces of sheaves over surfaces [AB13, ABCH13, Mea12]. [...]

An approach to complex interpolation of compact subsets of $\Bbb C^n$, including Brunn-Minkowski type inequalities for the capacities of the interpolating sets, was developed recently by means of plurisubharmonic geodesics between relative extremal functions of the given sets. Here we show that a much better control can be achieved by means of the geodesics between weighted relative extremal functions. In particular, we establish convexity properties of the capacities that are stronger than those given by the Brunn-Minkowski inequalities., Final version; to appear in Arnold Math. J

In this paper, we turn our attention to the mathematical model to simulate steady, hydromagnetic, and radiating nanofluid flow past an exponentially stretching sheet. A numerical modeling technique, simplified finite difference method (SFDM), has been applied to the flow model that is based on partial differential equations (PDEs) which is converted to nonlinear ordinary differential equations (ODEs) by using similarity variables. For the resultant algebraic system, the SFDM uses the tridiagonal matrix algorithm (TDMA) in computing the solution. The effectiveness of numerical scheme is verified by comparing it with solution from the literature. However, where reference solution is not available, one can compare its numerical results with the results of MATLAB built-in package bvp4c. The velocity, temperature, and concentration profiles are graphed for a variety of parameters, i.e., Prandtl number, Grashof number, thermal radiation parameter, Darcy number, Eckert number, Lewis number, and Brownian and thermophoresis parameters. The significant effects of the associated emerging thermophysical parameters, i.e., skin friction coefficient, local Nusselt number, and local Sherwood numbers are analyzed and discussed in detail. Numerical results are compared from the available literature and found a close agreement with each other. It is found that the Eckert number upsurges the velocity curve. However, the dimensionless temperature declines with the Grashof number. It is also shown that the SFDM gives good results when compared with the results obtained from bvp4c and results from the literature.

Every set $\Lambda\subset R$ such that the sum of $\delta$-measures sitting at the points of $\Lambda$ is a Fourier quasicrystal, is the zero set of an exponential polynomial with imaginary frequencies., Comment: 8 pages

Let $Q$ be an acyclic quiver and $w \geqslant 1$ be an integer. Let $\mathsf {C}_{-w}({\mathbf {k}} Q)$ be the $(-w)$-cluster category of ${\mathbf {k}} Q$. We show that there is a bijection between simple-minded collections in $\mathsf {D}^b({\mathbf {k}} Q)$ lying in a fundamental domain of $\mathsf {C}_{-w}({\mathbf {k}} Q)$ and $w$-simple-minded systems in $\mathsf {C}_{-w}({\mathbf {k}} Q)$. This generalises the same result of Iyama–Jin in the case that $Q$ is Dynkin. A key step in our proof is the observation that the heart $\mathsf {H}$ of a bounded t-structure in a Hom-finite, Krull–Schmidt, ${\mathbf {k}}$-linear saturated triangulated category $\mathsf {D}$ is functorially finite in $\mathsf {D}$ if and only if $\mathsf {H}$ has enough injectives and enough projectives. We then establish a bijection between $w$-simple-minded systems in $\mathsf {C}_{-w}({\mathbf {k}} Q)$ and positive $w$-noncrossing partitions of the corresponding Weyl group $W_Q$.

Sammendrag på norsk Denne avhandlingen omhandler geometriske og funksjonalanalytiske aspekter ved tid-frekvensanalyse. Mer spesifikt drøfter avhandlingen de tre følgende delvis relaterte temaene: Dekomponeringsrom: I artikkel A og artikkel B studerer vi dekomponeringsrom ved å bruke teknikker fra geometri i stor skala. Dekomponeringsrom er en klasse med funksjonsrom som inkluderer modulasjonsrommene og Besovrommene. Vi utvikler et begrep om geometriske avbildninger mellom forskjellige dekomponeringsrom. I tillegg utbygger vi teorien om dekomponeringsrom på nilpotente liegrupper. Hovedresultatet vårt i denne retningen etablerer at en stor klasse med modulasjonsrom på nilpotente liegrupper er ulik fra de euklidske modulasjonsrommene. Waveletrom: I artikkel C studerer vi waveletrom ved å bruke teknikker fra reproduserbar kjerne hilbertrom. Et spesialtilfelle av waveletrom har blitt undersøkt tidligere i tid-frekvensanalyse under navnet Gabor-rom. Vi oppdager en sammenheng mellom fullstendige interpolerende Gabor-rom og HRT-formodningen i tid-frekvensanalyse. Kvanteharmonisk analyse: I artikkelDog artikkel E utvikler vi kvanteharmonisk analyse på den affine gruppen. Dette krever en grundig undersøkelse av den affine Wigner-distribusjonen. Av spesiell interesse er definisjonen av tillatelige operatorer. Mange av resultatene vi gir sikter mot å knytte sammen den affine Weyl-kvantiseringen med konvolusjoner på den affine gruppen. Beskrivelsene ovenfor hentyder at avhandlingen setter søkelys på å generalisere tid-frekvensanalyse i forskjellige retninger. På tross av generaliteten i avhandlingen utvikler vi resultater som vi tror er av interesse for konkrete eksempler. En rød tråd gjennom avhandlingen er unikhet: Artikkel A: Vi undersøker forskjellige egenskaper fra geometri i stor skala (f.eks. asymptotisk dimensjon og hyperbolskhet) som differensierer forskjellige dekomponeringsrom. Slike egenskapene bestemmer hvorvidt forskjellige dekomponeringsrom kan bli kontinuerlig avbildet inn i hverandre slik at geometriske egenskaper blir bevart. Artikkel B: Vi konstruerer modulasjonsrom på bestemte nilpotente liegrupper. Av særskilt interesse er spørsmålet om hvorvidt de nye funksjonsrommene er ulike fra de klassiske euklidske modulasjonsrommene. Vi besvarer dette spørsmålet bekreftende. Artikkel C: Vi undersøker waveletrom og deres egenskaper som reproduserbar kjerne hilbertrom. En konsekvens er at vi utvider tidligere resultater om unikhet av waveletrom ved å bruke verktøy fra representasjonsteori. Artikkel D: Vi betrakter den affineWigner-distribusjonen og dens grunnleggende egenskaper. En av anvendelsene er et minimeringsproblem for den affine Wigner-distribusjonen. Selv for den tradisjonelle Wigner-distribusjonen er det uklart hvor mange unike minimerere som eksisterer. Vi besvarer dette spørsmålet for Wigner-distribusjonen og den affine Wigner-distribusjonen. Artikkel E: Vi utvikler et kvanteharmonisk analyse rammeverk for den affine gruppen. Rammeverket benytter seg av en operator som kalles den affine paritetsoperatoren. Det viser seg at vår fremgangsmåte er unik fra tidligere fremgangsmåter i den forstand at vi kan representere kvantiseringen som operatorkonvolusjon med den affine paritetsoperatoren. Abstract in English This thesis revolves around both geometric and functional analytic aspects of timefrequency analysis. More specifically, the thesis deals with the following three related topics: Decomposition Spaces: Both Paper A and Paper B study decomposition spaces through the lens of large scale geometry. Decomposition spaces include the modulation spaces and the Besov spaces as special cases. We develop a notion of geometric embeddings between different decomposition spaces in Paper A. In Paper B we advance the theory of decomposition spaces on nilpotent Lie groups. Our main result in this direction establishes that a large class of modulation spaces on nilpotent Lie groups is distinct from their Euclidean counterparts. Wavelet Spaces: In Paper C we study wavelet spaces by utilizing techniques from reproducing kernel Hilbert spaces. A special case of wavelet spaces has been investigated in time-frequency analysis under the name Gabor spaces. We discover a connection between fully interpolating Gabor spaces and the HRT-conjecture in time-frequency analysis. Quantum Harmonic Analysis: In Paper D and Paper E we develop quantum harmonic analysis on the affine group. This requires a careful examination of the affineWigner distribution and the affineWeyl quantization. Of particular interest is the development of a notion of admissibility for operators in the affine setting. Many of our results are aimed at connecting the affine Weyl quantization with convolutions on the affine group. As indicated by the descriptions above, the thesis is concerned with generalizing time-frequency analysis in various directions. Despite the general approach considered in the thesis, some of the developed results are new even in well-studied settings. A common conceptual theme across the papers is distinctness: Paper A: We consider various properties from large scale geometry (e.g. asymptotic dimension and hyperbolicity) that allows us to distinguish different decomposition spaces. These properties even determine whether different decomposition spaces can embed into one another while preserving geometric properties. Paper B: We construct modulation spaces on certain nilpotent Lie groups. Of central importance is the question of whether these new function spaces are distinct from the classical Euclidean modulation spaces. We answer this question affirmatively. Paper C: We investigate wavelet spaces and their properties as reproducing kernel Hilbert spaces. One side-effect is thatwe generalize previously known results regarding distinctness of wavelet spaces by using well-known tools from representation theory. Paper D: We consider the affine Wigner distribution and its basic properties. Among the applications is a minimization problem for the affine Wigner distribution. Even for the standard Wigner distribution it is not clear how many distinct minimizers exist. We settle this question in both the Heisenberg and the affine setting. Paper E: We develop a quantum harmonic analysis framework for the affine group. Our framework heavily uses the so-called affine parity operator. It turns out that our approach is distinct from previous affine quantizations in the literature. In particular, we can represent the quantization procedure as operator convolutions with the affine parity operator.

In factor analysis and structural equation modeling non-normal data simulation is traditionally performed by specifying univariate skewness and kurtosis together with the target covariance matrix. However, this leaves little control over the univariate distributions and the multivariate copula of the simulated vector. In this paper we explain how a more flexible simulation method called vine-to-anything (VITA) may be obtained from copula-based techniques, as implemented in a new R package, covsim. VITA is based on the concept of a regular vine, where bivariate copulas are coupled together into a full multivariate copula. We illustrate how to simulate continuous and ordinal data for covariance modeling, and how to use the new package discnorm to test for underlying normality in ordinal data. An introduction to copula and vine simulation is provided in the appendix.

We study the Newman--Unti (NU) group from the viewpoint of infinite-dimensional geometry. The NU group is a topological group in a natural coarse topology, but it does not become a manifold and hence a Lie group in this topology. To obtain a manifold structure we consider a finer Whitney-type topology. This turns the unit component of the NU group into an infinite-dimensional Lie group. We then study the Lie theoretic properties of this group. Surprisingly, the group operations of the full NU group become discontinuous, whence the NU group does not support a Lie group structure. The NU group contains the Bondi--Metzner--Sachs (BMS) group as a subgroup, whose Lie group structure was constructed in a previous article. It is well known that the NU Lie algebra splits into a direct sum of Lie ideals of the Lie algebras of the BMS group and conformal rescalings of scri. However, the lack of a Lie group structure on the NU group implies that the BMS group cannot be embedded as a Lie subgroup therein., 37 pages, article; minor revisions; version to appear in Classical and Quantum Gravity

In the present work, we investigate the annihilation of persistent localized activity states (bumps) in a Wilson-Cowan type two-population neural field model in response to $\alpha $ -type spatio-temporal external input. These activity states serves as working memory in the prefrontal cortex. The impact of different parameters involved in the external input on annihilation of these persistent activity states is investigated in detail. The $\alpha $ -type temporal function in the external input is closer to natural phenomenon as observed in Roth et. al . ( Nature Neuroscience , vol. 19 (2016), 229–307). Two types of eraser mechanism are used in this work to annihilate the spatially symmetric solutions. Initially, if there is an activity in the network, inhibitory external input with no excitatory part and over excitation with no inhibition in the external input can kill the activity. Our results show that the annihilation of persistent activity states using $\alpha $ -type temporal function in the external input is more roubust and more efficient as compare to triangular one as used by Yousaf et al. ( Neural networks. , vol. 46 (2013), pp. 75–90). It is also found that the relative inhibition time constant plays a crucial role in annihilation of the activity. Runge-Kutta fourth order method has been employed for numerical simulations of this work.

MIXTURE THEORY has been developed based on balance laws and conservation principles which is well known in continuum mechanics, and has been widely applied to systems which can be characterized as a mixture of interacting continua. In this thesis we present a general mathematical model for the compressible and incompressible multiphase flows based on the theory of mixtures and investigate the corresponding numerical schemes that are used to solve different kinds of flow phenomena. Mathematical analysis, which essentially boils down to deriving various estimates in appropriate functional spaces, is also used as a tool to gain insight into the models that have been explored. The first part (Paper I and II) of this thesis implements a novel two-phase momentum-equation-approach to include the viscous coupling effect involved in counter-current and co-current flows during spontaneous imbibition process for oil recovery. The formulation can automatically capture the effective relative permeabilities of the flowing phases and offers improvements over conventional modelling. The model also accounts for the fact that oil must overcome the so-called capillary backpressure when it is produced counter-currently. It is shown that this parameter can influence the extent of counter-current production and hence viscous coupling. The second part (i.e., Paper III-V) of this thesis focuses on the development of numerical schemes for solving the general multiphase flow model. We consider two-phase and three-phase flow in porous media for both incompressible and compressible cases. We also explore in Paper III and IV different estimates that can deepen the insight into the system which is studied as well as ensure existence of solutions subject to appropriate assumptions. Specifically, in Paper III, a general two-phase model without source terms is solved and the investigations indicate that interstitial velocity seems more natural to use in the viscous term than Darcy velocity for the momentum balance equation. In Paper IV, source terms are taken into account and different physical effects are highlighted such as compressibility and viscous coupling. Various a priori estimates are derived that give rise to an existence result for the general two-phase model with and without source terms. In addition, we also extended the two-phase model to the three-phase model in Paper V and illustrated the stability of the developed numerical schemes by investigating several numerical examples, both for the case with incompressible and compressible fluids. In the last part of this thesis, represented by Paper VI, we explore a general cell-fluid Navier-Stokes model to mimic an evolution process of the bacterial plumes development. The generality of the mixture theory approach allows us to incorporate a new migration effect, so-called chemotaxis, in the momentum balance for the cell phase. Chemotaxis and gravity segregation are two main driving forces that result in a complex flow pattern formation of bacterial. The nonlinear dynamics is explored in a 2D setting by using an appropriate finite difference scheme. This model can be interpreted as a generalization of the well-known chemotaxis-Stokes model which has attracted much attention the last decade.

This chapter is about a change in geometry education that took place in the last century. We discuss the emergence of meaningful geometry in the Netherlands. Of course, this was not an isolated reform. Worldwide, mathematicians and mathematics educators came up with new ideas as an alternative for the traditional axiomatic approach to teaching geometry. Already at the end of the 19th century, Klein had made a start with this by advocating a transformation geometry, but in this approach the axiomatic structure still played a main role for ordering activities. This was not the case in the work of Fröbel and Montessori who by building on students’ intuitions and their attention for students’ development of spatial insight were important driving forces towards a meaningful approach to geometry education. In the Netherlands, the pioneers of such a geometry were Tatiana Ehrenfest and Dieke van Hiele–Geldof. Freudenthal was a great promoter of their ideas. For him, geometry is ‘grasping space’, meaning that geometrical experiences should start with the observation of phenomena in reality. Supported by Freudenthal, from the 1970s on, experiments were carried out in the Netherlands to develop a new intuitive and meaningful approach to geometry education, in which the focus was on spatial orientation. How big the change in geometry education that resulted from these experiments was, is illustrated in this chapter by comparing geometry problems from two Dutch mathematics textbooks: one from 1976 and one from 2002.

In this study we investigated the structure of quantitative competence of kindergartners by testing a hypothesized four-factor model of quantitative competence consisting of the components counting, subitizing, additive reasoning and multiplicative reasoning. Data were collected from kindergartners in the Netherlands (n = 334) and in Cyprus (n = 304). A confirmatory factor analysis showed that the four-factor structure fitted the empirical data from the Netherlands. For the Cyprus data a one-factor structure was found to have a more adequate fit. Regarding the effect of country on performance, a comparison at item level showed that the kindergartners in the Netherlands outperformed those in Cyprus in the majority of quantitative competence items. Analyses of variance revealed for each country a significant effect of kindergarten year on performance, with children in K2 (second kindergarten year) outperforming those in K1 (first kindergarten year). A statistical implicative analysis at item level revealed that in both countries the relevant implicative chain, showing what successful solving of an item implies for correct solving of another item, reflects by and large the sequential steps mostly followed in teaching kindergartners early number. This sequence starts with counting and subitizing, then continues with additive reasoning and finally multiplicative reasoning. These implicative chains also clearly show that the development of early quantitative competence is not linear. There are many parallel processes and cross-connections between the components of quantitative competence.

This chapter contains an overview of the most important textbook series used in the Netherlands from 1800 to 2010. We distinguish five time periods, and for each period we highlight the textbook series that are most characteristic. To describe the textbooks that were in fashion in the successive periods we distinguish three categories of textbooks: procedural, conceptual, and dual textbooks. The dual textbooks have elements of the first two. For the procedural textbook series, which are also referred to as ‘mechanistic’, memorisation of mathematical facts, automatisation on of operational procedures and recognising types of problems are the primary interest. Application is only considered at the very end of the teaching trajectory, and then rarely. Smart, flexible (mental) calculations and estimating are not part of the program. The conceptual textbook series have an opposite approach. In learning mathematical facts and procedures, understanding is highly valued, and applications are included from the start as the basis for this. Number sense, flexible (mental) calculation, and estimation are central, next to algorithmic calculation. Students can design their own problems, develop solution methods and work on their own level. As expected, using different textbook series with different content and teaching methods results in pursuing different goals in mathematics education, which in turn results in different learning outcomes, as has been shown by national evaluations of progress in educational achievement.

In this introductory chapter I give a preview of the landscape of issues concerning mathematics education in the Netherlands and the role of Realistic Mathematics Education (RME) that one can come across in this volume, which contains the reflections of twenty-eight Dutch mathematics didacticians on teaching and learning mathematics in the Netherlands. Although all chapters have their own focus and mostly only discuss one particular aspect, together they provide a rich inside view into what is worth knowing of Dutch mathematics education and RME. The preview highlights some significant topics from these chapters, such as what tasks are preferred in RME to elicit students’ mathematical thinking, RME’s focus on the usefulness of mathematics, the role of common sense and informal knowledge, changes over time in the content of the mathematics curriculum, aspects of the Dutch educational system, including teacher education and assessment, the implementation of RME, and the context of developing RME.

This paper expands further on a category theoretical formulation of Hochschild cohomology for monoid objects in monoidal categories enriched over abelian groups, which has been studied in arXiv:1605.00842. This topic was also presented at ISCRA, Isfahan, Iran, April 2019. The present paper aims to provide a more intuitive formulation of the Hochschild cochain complex and extend the definition to Hochschild cohomology with values in a bimodule object. In addition, an equivalent formulation of the Hochschild cochain complex in terms of a cosimplicial object in the category of abelian groups is provided., Comment: 12 pages

In a three-phase study, with a total of 40 third-grade teachers and their 830 students, teachers were supported to use classroom assessment techniques (CATs) to reveal their students’ knowledge of number operations. In phase I, four teachers and 66 third-grade students participated in five monthly workshops in which CATs were co-designed and their use was discussed. In phase II, the first phase was replicated with four workshops with six different teachers and 148 third-grade students. In these two exploratory phases, we evaluated student achievement on a standardized national mathematics test in a pre-/posttest design and compared changes herein to changes in the national norm sample. In phase III, a control condition was added to the design to experimentally investigate the effect on student achievement with 30 teachers and 616 third-grade students. Teachers were randomly assigned to participate in 0, 1, 2, or 3 1-hour workshops. In all three phases, we found a significant increase in students’ mathematics achievement scores on the standardized mathematics test. In phase III, the increase was significantly larger in the classes of teachers participating in three workshops than in classes with less workshops. Additionally, results from the analysis of classroom observations, feedback forms, and interviews indicate that teachers could easily integrate the CATs into their practice and could gather valuable information on their students. The results from the different phases of this study combined indicate that supporting teachers in their development and use of classroom assessment in mathematics may contribute to the improvement of students’ mathematics achievement.

Reasoning about graphical representations representing dynamic data (e.g., distance changing over time), including interpreting, creating, changing, combining, and comparing graphs, can be considered a domain-specific operationalization of the general twenty-first century skills of creative, critical thinking and solving problems. This paper addresses the issue of how these 21st century skills of interpreting and creating graphs can be supported in a six-lesson teaching sequence about graphing motion. In this teaching sequence, we focused on the potential of an embodied learning environment to facilitate the development of primary school students’ reasoning about motion graphs by having primary school students (9–11 years) ‘walk’ graphs in front of a motion sensor to generate distance-time graphs. We asked: How does students’ reasoning about graphing motion develop over a six-lesson teaching sequence within an embodied learning environment? Based on the collected data, we examined changes in students’ level of reasoning on graph interpretation and graph construction tasks using a repeated measurement design. Additionally, we present two teaching episodes showing instances of how perceptual-motor experiences during the lessons aided students’ reasoning about graphical representations of motion. Results show that students went from iconic understanding towards understanding in which they reasoned based on one or two variables when interpreting and constructing graphical representations of motion events. At these higher levels of reasoning these students showed understanding of modelling motion in line with the intended 21st century skills of generating, refining, and evaluating graphs. Paid Open Access

This chapter is a synthesis of visions on and experiences with Realistic Mathematics Education (RME) described in the eighteen following chapters of this volume by forty-four authors from fifteen different countries. Through a process of synthesizing information from these chapters and combining and contrasting what the authors wrote about RME, a comprehensive image emerged of the theory and practice of RME, together with some new vistas. The chapter is structured around the following themes: making acquaintance with RME, narratives of first experiences with RME, highlighted outstanding features of RME, processes of implementation of RME and their challenges, adaptations of RME, criticisms of RME, and the flavours of RME that can be found in foreign curricula, textbooks, instructional materials, and teaching methods. Finally, to conclude the chapter, I reflect on new insights related to RME and directions for its further development that can be gained from this input from abroad.

This dissertation addresses the applications and challenges of both laboratory experiments and mathematical modeling at different scales, where the main character is biofilm. Thus, the first part of this work shows biological, chemical and physical concepts for the laboratory experiments and mathematical terms for the modeling, upscaling and numerical solutions. The second part contains the research papers. In our research, we are interested in studying the biofilm to improve the oil extraction. Most of the biofilm models are based on simplifying assumptions, e.g. impermeability, a constant biofilm density and accounting for diffusion but neglecting convection for transport of nutrients. In this work, we propose a pore-scale model for a permeable multi-component biofilm including a variable biofilm density, detachment and transport of nutrients due to convection and diffusion. It is through laboratory experiments that we identify the key processes and variables that need to be considered. Accordingly, we use experimental determined parameters and compute some of the parameters through calibration. In addition, we study the sensitivity of the parameters in the mathematical models. Pore-scale models are important because they aim to describe physical phenomena in detail and one can derive core-scale models through upscaling. Then, we can reflect the effects of the pore-scale processes on the core scale. Upscaling of pore-scale models allows us to describe the average behavior of a system in an accurate manner with relatively low computational effort. Then, we upscale this pore-scale model in two different geometries: a thin channel and a thin tube, in order to derive one-dimensional effective equations, by investigating the limit as the ratio of the aperture to the length approaches to zero. In the core-scale laboratory experiments, biofilm is grown in cylindrical cores. Permeability and porosity changes over time at different flow rates and nutrient concentrations are studied. Numerical simulations are performed to compare with the experimental results. We also present how to extend the model to include chemotaxis and interfacial tension reduction due to surface active compounds. Mathematical models for biofilms are based on coupled non-linear partial differential equations and ordinary differential equations, which may be challenging to solve. Therefore, it is necessary to use advanced numerical methods and simulations to predict the behavior on time of the unknowns in these complex systems. We present some of the common space discretizations, time discretizations and numerical solvers for these models. We also discuss the difficulty of free boundary problems and the numerical techniques to deal with them. Last but not least, we discuss the challenges of parameter estimation and the application of sensitivity analysis.

In this paper, we describe four studies that have been done in the project Improving Classroom Assessment in China (ICA-C). The project was a sequel to the ICA project in the Netherlands. To shed light on the current situation of Chinese primary school mathematics teachers’ perception and practice of classroom assessment, we first did a review study based on teacher-written papers addressing classroom assessment. Then a large-scale questionnaire survey was conducted by means of which we could identify different assessment profiles of teachers. With the gained knowledge, two other studies were set up to explore the possibility for improving Chinese primary school mathematics teachers’ assessment activities, with a particular focus on using classroom assessment techniques in the domain of division and multiplication.

We prove that if an algebra is either selfinjective, local or graded, then the Hochschild homology dimension of its trivial extension is infinite. © 2017. This is the authors' accepted and refereed manuscript to the article. The final authenticated version is available online at: http://www.ams.org/journals/proc/2017-145-04/S0002-9939-2016-13363-7/home.html

In a two-period game, Player 1 produces zero-day exploits for immediate deployment or stockpiles for future deployment. In Period 2, Player 1 produces zero-day exploits for immediate deployment, supplemented by stockpiled zero-day exploits from Period 1. Player 2 defends in both periods. The article illuminates how players strike balances between how to exert efforts in the two periods, depending on asset valuations, asset growth, time discounting, and contest intensities, and when it is worthwhile for Player 1 to stockpile. Eighteen parameter values are altered to illustrate sensitivity. Player 1 stockpiles when its unit effort cost of developing zero-day capabilities is lower in Period 1 than in Period 2, in which case it may accept negative expected utility in Period 1 and when its zero-day appreciation factor of stockpiled zero-day exploits from Period 1 to Period 2 increases above one. When the contest intensity in Period 2 increases, the players compete more fiercely with each other in both periods, but the players only compete more fiercely in Period 1 if the contest intensity in Period 1 increases.

This study investigated whether providing opportunity-to-learn can improve Indonesian students’ performance in solving context-based mathematics tasks. On the basis of an inventory of Indonesian students’ difficulties with these tasks and an analysis of textbooks and classroom practices, an intervention program for mathematics teachers was developed. This program contained tasks with relevant and essential contexts with missing or superfluous information, but without explicitly given mathematical procedures. The program also comprised guidelines for a consultative teaching approach with metacognitive prompts and questions for discussion to promote reflection in class. A field experiment with a pretest-posttest control-group design was carried out in six junior high schools in Indonesia involving 299 eight-graders. Students in the experimental group made significantly more progress on solving context-based mathematics tasks than students in the control group. Furthermore, an analysis of students’ errors revealed that experimental students made significantly fewer task comprehension errors than control students. These results show that providing opportunity-to-learn, that is offering context-based tasks to students, which require mathematical modeling, and having teachers knowing the characteristics of such tasks and using a consultative teaching approach, can improve students’ ability in solving context-based tasks.

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I denne oppgaven har jeg analysert tre matematikklæreverk for første og andre trinn på barneskolen fra de største forlagene i Norge, med hovedfokus på hvordan disse læreverkene formidler begreper og hvordan dette i sin tur kan påvirke elevenes matematikkforståelse på disse trinnene. Undervisningen i hvert enkelt klasserom vil selvsagt være veldig forskjellig ettersom den planlegges og gjennomføres av forskjellige lærere med forskjellig utgangspunkt. Når det er sagt, vet vi likevel at lærebokas rolle i matematikkfaget er stor. Omfattende studier har vist at mange matematikklærere legger hele sin undervisning opp etter læreverkene, med den konsekvens at læreboka også står ansvarlig for en stor del av hvordan matematikk formidles til elevene. Analysen tar utgangspunkt i Systematisk Begrepsundervisning etter Magne Nyborg, og da særlig Begrepsundervisningsmodellen (BU-modellen). BU-modellens tre faser; assosiasjonsfasen, diskriminasjonsfasen og generaliseringsfasen gjenspeiler de naturlige prosessene ved dannelsen av et nytt begrep, relativt uavhengig av hvem det er som skal lære begrepet. Denne forskningsbaserte undervisningsmodellen har gjennom omfattende studier vist seg valid med tanke på å fremme begrepsforståelse hos et bredt spekter av målgrupper, blant annet undervisning av 6-åringer i førskole og de første trinn på barneskolen, som også er målgruppen for de analyserte læreverkene. In this study, I have analysed mathematical textbooks from Norway's largest publishers for the first and second year of primary school. The main focus was on how these books convey concepts and how this, in turn, can affect the pupils' mathematical understanding. How mathematics is taught differs greatly, due to differences between individual teachers and classrooms. Having said that, we still know that the textbook's role in mathematics is significant. Many maths teachers rely entirely on the textbooks and plan their lessons accordingly. For some pupils, the textbook therefore becomes the most important source of mathematical knowledge. The analyses are based on Magne Nyborgs theory about systematically teaching of concepts, and especially the Concept Teaching Model (CTM). The three-phase CTM consists of the association phase, the discrimination phase and the generalization phase. reflects the natural processes of the formation of a new concept, independent of who is learning the concept. Through this study, this research-based teaching model has proven to be valid in promoting conceptual understanding to a wide range of target groups, including education of 6-year-olds in preschool and the first stages of primary school, which is also the target group for the analysed textbooks. M-LUN

The Unsaturated Flow In Deformable Porous Media (UFIDPM) plays a crucial role in several academic and industrial applications such as; cracks induced by desiccation, collapsing soils, ground movement involving expansive soils, lateral earth surfaces, stability of vertical excavations, natural slopes subjected to environmental changes, construction and operation of a dam, etc. Typically, studying these systems with experimental techniques is impractical. On the other hand, numerical simulations allow us to investigate several scenarios in a short time and with reduced costs. At the moment, only Finite Element Method (FEM) based codes were available. FEM often shows conservative issues and when is applied to UFIDPM instabilities in the limit of incompressibility have been reported. Due to this issues, we were motivated to develop the first Cell-Centered Finite Volume Method (CCFVM) based code for solving UFIDPM. The governing equations are derived following the extended Biot’s theory of three-dimensional consolidation which results in a coupled hydro-mechanical system. For the flow problem, we use the Richards’ equation whereas the mechanical problem is modeled using the linear elasticity equations. For the spatial discretization, we use multi-points approximations schemes. Specifically, Multi Point Flux Approximation (MPFA) for the flow problem and Multi Point Stress Approximation (MPSA) for the mechanical problem. Moreover, the time discretization of the equations was obtained using Backward Euler (BE). The code was implemented in MATLAB R2017b were two core toolboxes (MRST and FV-BIOT) were used. The resulting non-linear set of equations was solved using the Newton method together with Automatic Differentiation (AD). To test the capability of the code several sub-problems were validated. Furthermore, we present a numerical application where we focus on the desiccation process of a clayey soil in a Petri-dish. In this experiment, the water content reduction is caused by instantaneous water evaporation controlled by atmospheric conditions. Finally, by carefully post-processing the resulting stress field, the zones of tensile stress concentration, which corresponds to the areas where cracks are more likely to initiate, are identified. Master's Thesis in Mathematics MAMN-MAB MAB399

Quadratic differentials first appeared in 1930s in works of Teichmuller in connection with moduli problem for Riemann surfaces. Later it was revealed that quadratic differ- entials and their trajectories give solutions to extremal problems for moduli of families of curves and extremal partition of Riemann surfaces. Over the last decades there has been discovered a connection between quadratic differentials and extremal problems of different nature related to minimal surfaces, potential theory, approximation theory, mathematical physics. In this work, we study quadratic differentials, discover new properties of certain types of quadratic differentials and explore the connection of quadratic differentials with other disciplines, such as discrete mathematics, topology and even applied math- ematics. We apply the theory of quadratic differentials to solve problems of complex analysis and study analytical problems that are connected with quadratic differentials. The dissertation is organized as follows. Chapter 1 contains an introduction to the notions and facts that are necessary for understanding the main results of this work. Section 1.1 gives a short introduction to quadratic differentials and their trajectory structure. It also recalls the definition of reduced moduli of digons. Section 1.2 de- scribes the tools from discrete mathematics and topology that are used in this work. Chapter 2 gives a summary of the research papers that constitute the main scientific contributions of the thesis. Both Chapter 1 and Chapter 2 are followed by lists of refer- ences. The following papers are included in Chapter 3. Paper A: A. Frolova, M. Levenshtein, D. Shoikhet, and A. Vasil’ev, Boundary distortion estimates for holomorphic maps, InComplex Analysis and Operator Theory, 8 (2013), no. 5, 1129–1149. Paper B: A. Frolova, A. Vasil’ev, Combinatorial description of jumps in spectral networks, accepted to Proceedings AMS. Paper C: A. Frolova, D. Khavinson, and A. Vasil’ev, Polynomial lemniscates and their fingerprints: from geometry to topology, Submitted to New Trends in Complex and Harmonic Analysis. Paper D: A. Frolova, M. Solberg, and A. Vasil’ev, Pure braids and homotopy classes of closed loops, paper in preparation. In Paper A we apply extremal partitions and moduli of digons to obtain a boundary distortion estimate for a conformal self-mapping of the unit disk with two fixed points on the boundary. Paper B studies combinatorial structure of the family of quadratic differentials with a finite simple pole that possess a short trajectory. In Paper C we discover some properties of fingerprints of polynomial lemniscates and construct a non-unitary operad that realizes a process of approximation of two dimensional shapes by polynomial lemniscates. In Paper D we define an action of a braid group on homotopy classes of closed curves on a punctured sphere, which is related to a problem of constructing an operad on quadratic differentials with several double poles.

The so-called Whitham equation arises in the modeling of free surface water waves, and combines a generic nonlinear quadratic term with the exact linear dispersion relation for gravity waves on the free surface of a fluid with finite depth. In this work, the effect of incorporating capillarity into the Whitham equation is in focus. The capillary Whitham equation is a nonlocal equation similar to the usual Whitham equation, but containing an additional term with a coefficient depending on the Bond number which measures the relative strength of capillary and gravity effects on the wave motion. A spectral collocation scheme for computing approximations to periodic traveling waves for the capillary Whitham equation is put forward. Numerical approximations of periodic traveling waves are computed using a bifurcation approach, and a number of bifurcation curves are found. Our analysis uncovers a rich structure of bifurcation patterns, including subharmonic bifurcations, as well as connecting and crossing branches. Indeed, for some values of the Bond number, the bifurcation diagram features distinct branches of solutions which intersect at a secondary bifurcation point. The same branches may also cross without connecting, and some bifurcation curves feature self-crossings without self-connections. This is a submitted manuscript of an article published by Elsevier Ltd in Physica D: Nonlinear Phenomena, 6 December 2016.

Recently, M.~Bouafoa, et al. investigated a new kernel function which differs from the self-regular kernel functions. The kernel function has a trigonometric Barrier Term. In this paper we generalize the analysis presented in the above paper for $P_{*}(\kappa)$ Linear Complementarity Problems (LCPs). It is shown that the iteration bound for primal-dual large-update and small-update interior-point methods based on this function is as good as the currently best known iteration bounds for these type methods. The analysis for LCPs deviates significantly from the analysis for linear optimization. Several new tools and techniques are derived in this paper.

Author's accepted version (post-print). This is a post-peer-review, pre-copyedit version of an article published in São Paulo Journal of Mathematical Sciences. The final authenticated version is available online at: http://dx.doi.org/10.1007/s40863-016-0046-4. This paper is a survey on some of the most basic results in the theory of quasi-hereditary algebras. In the last section we briefly discuss a recent development.

A generic strictly semistable bundle of degree zero over a curve \(X\) has a reducible theta divisor, given by the sum of the theta divisors of the stable summands of the associated graded bundle. The converse is not true: Beauville and Raynaud have each constructed stable bundles with reducible theta divisors. For \(X\) of genus \(g \ge 5\), we construct stable vector bundles over \(X\) of rank \(r\) for all \(r \ge 5\) with reducible and nonreduced theta divisors. We also adapt the construction to symplectic bundles. In the “Appendix”, Raynaud’s original example of a stable rank 2 vector bundle with reducible theta divisor over a bi-elliptic curve of genus 3 is generalized to bi-elliptic curves of genus \(g \ge 3\).

In this study, we explored the insights that Chinese primary mathematics teachers gained into their students&rsquo, mathematical understanding from using classroom assessment techniques (CATs). CATs are short teacher-initiated targeted assessment activities proximate to the textbook, which teachers can use in their daily practice to make informed instructional decisions. Twenty-five third-grade teachers participated in a two-week program of implementing eight CATs focusing on the multiplication of two-digit numbers, and filled in feedback forms after using the CATs. When their responses described specific information about their students, emphasized the novelty of the gained information, or referred to a fitting instructional adaptation, and these reactions went together with references to the mathematics content of the CATs, the teachers&rsquo, responses were considered as evidence of gained insights into their students&rsquo, mathematics understanding. This was the case for three-quarters of the teachers, but the number of gained insights differed. Five teachers gained insights from five or more CATs, while 14 teachers did so only from three or fewer CATs, and six teachers showed no clear evidence of new insights at all. Despite the differences in levels of gained insights, all the teachers paid more attention to descriptions of students&rsquo, performance than to possible instructional adaptations.

In today's world, oil remains the main source of energy. After discovering a petroleum reservoir, one can extract about 30% of the oil by using and maintaining the initial pressure in the reservoir through water ooding (first and second phase oil recovery). Nevertheless, 60-70% of oil remains in the reservoir after this, so called conventional recovery, so enhanced oil recovery (EOR) is strongly needed. Microbial enhanced oil recovery (MEOR) is an environmentally friendly and very promising EOR technology. In this thesis we implement a non-standard model for MEOR that includes the interfacial area between the oil and water. We consider the two-phase ow equations and multicomponent reactive transport equations for three components: bacteria, nutrients and biosurfactants. The growth of bacteria affects the properties of the medium (porosity, permeability), up to the extreme case of pore clogging (bioclogging) and generation of new paths. Biosurfactants lower the interfacial tension, which improves the oil recovery. We consider in this thesis, for the fi The motivation to include the interfacial area in the model is to eliminate the hysteresis in the capillary pressure relationship and model that biosurfactants are mainly living at the oil-water interface. A typical mathematical model for MEOR consists on nonlinear coupled partial differential equations (PDEs) and ordinary differential equations (ODEs). The spatial discretization is obtained using finite differences (FD) and two-point ux approximation (TPFA), and the time discretization using backward Euler (BE). We make the implementation for 1-D and 2-D domains using cell-centered grids. We present in detail the discretization of the equations, including the treatment of the boundary conditions. After the discretization in space and time, the problems to be solved at each time step are still nonlinear. For solving these equations, we use an implicit scheme that considers a linear approximation of the capillary pressure gradient, which results in an efficient and stable scheme. The code for running the simulations is written in MATLAB. Following this, we test the code with analytic solutions and benchmark simulations. We consider a 1-D porous medium where we study the spatial distribution and the evolution in time of the average pressure, water saturation, oil-water interfacial area, capillary pressure, porosity, permeability ratio, residual oil water saturation and bacterial, nutrient and biosurfactant concentrations. After, we make a sensitivity analysis in order to examine the effects of the relevant model parameters. Finally, we make simulations considering a porous medium with a thief zone, which is a 2-D problem. A comprehensive model, including two-phase ow, bacteria, nutrients and biosurfactants was implemented. The model has been tested using analytic and benchmark problems. For the first time, the role of interfacial area in MEOR was studied. We showed that different predictions of oil recovery are obtained by including the availability of interfacial area in the model. Nevertheless, it is necessary to do more experiments in the laboratory in order to compare with the numerical simulations and validate the model assumptions. MAMN-MAB MAB399

The common theme in this thesis is the study of constructive provability: in particular we investigate aspects of finite sets and Kan simplicial sets from a constructive perspective. There are numerous definitions of finiteness which are classically equivalent but not constructively so. In other words, constructive mathematics is able to distinguish between more notions of finiteness. We start by investigating some relationships between several ways of defining finiteness for sets of natural numbers. As a new result, we give strictly bounded a precise placement in a hierarchy of definitions of finiteness. We also investigate streamless sets, which constitutes another notion of finite- ness. Streamless sets require neither decidable equality nor that the set is a subset of an enumerable set, and they are as such more general than strictly bounded sets. It is an open problem whether the Cartesian product of two streamless sets is itself streamless. We show that this holds if at least one of the sets has decidable equality or is of bounded size. The problem remains open for the case where both streamless sets have undecidable equality and fail to be of bounded size. We also show that—in certain constructive systems—the addition of function extensionality makes equality within streamless sets decidable. Another notion of finiteness is Noetherian. Both streamless and Noetherian can be generalized to properties of binary relations, whereby such sets are those where equality is respectively streamless or Noetherian. We provide a proof that all Noetherian relations are streamless—notably, in a type system without inductively defined equality. This result immediately entails that all Noetherian sets are streamless within that type system. We proceed to investigate aspects of Kan simplicial sets, a notion coming from topology. Kan simplicial sets have recently caught the eye of the type theory community since they can be used to build models of Martin-L¨of type theory that validate the Univalence axiom. All known proofs of the following well-known theorem use classical logic: if simplicial sets X and Y are Kan simplicial sets then Y X is also a Kan simplicial set. This theorem plays an important role in the Kan simplicial set model of type theory. We investigate whether this theorem also holds constructively. The classical definition of the Kan property has at least two non-equivalent constructive interpretations, and we provide countermodels showing the constructive non-provability of the classical theorem above for both of these definitions of Kan simplicial sets.

Denne masteroppgåva handlar om korleis elevar på vidaregåande trinn – yrkesfagleg retning forstår omgrep innafor sannsyn. Føremålet har vore å avdekkja kva oppfatningar og misoppfatningar elevane har og kva løysingsstrategiar elevane nyttar når dei svarar på spørsmåla. Vidare har eg undersøkt i kva grad funna mine samsvarar med tidlegare funn som er referert i litteraturen i tillegg til å sjå nærare på korrelasjon mellom karakter og resultat. I studien er det nytta mixed methods for datainnsamlinga. Datainnsamlinga vart gjennomført i januar og februar 2015. Det vart først utført ei kvantitativ spørjeundersøking der 93 elevar deltok. Om lag 2/3 av desse går på vg1 i ulike yrkesfaglege retningar medan om lag 1/3 går på påbygg til generell studiekompetanse. Etterpå vart det utført ein kvalitativ studie der 9 elevar vart intervjua. Ingen av elevane hadde gjennomgått opplæring i sannsyn rett før datainnsamlinga. Spørsmåla byggjer i stor grad på spørsmål frå tidlegare undersøkingar slik at dei til ei viss grad kan seiast å vera kvalitetssikra. Alle misoppfatningane som eg ynskte å testa, synte seg å vera til stades i varierande grad hjå elevane. Dei talmessige resultata frå undersøkinga mi viser elles at det i stor grad er samanfallande med det forskarar har funne tidlegare, trass i at undersøkingane har stor variasjon i alder, skulebakgrunn og verdsdeler. Elevane nytta ulike løysingsstrategiar som vi og finn att i litteraturen og tidlegare forsking. Dette gjeld særleg bruk utfallstilnærming og 50-50 tilnærming. Gjennom intervjua fekk eg og avdekka at elevane gjerne stolar på intuisjonen sin og det er først når dei må tenkja i gjennom kva som står i spørsmålet, at dei kan gje eit normativt riktig svar. Det er særleg innafor sannsyn at bruk av intuisjon leier oss inn i ei misoppfatning. Vi nyttar som oftast det vi har lært tidlegare. MAUMAT 650

We generalize the total variation restoration model, introduced by Rudin, Osher, and Fatemi in 1992, to matrix-valued data, in particular, to diffusion tensor images (DTIs). Our model is a natural extension of the color total variation model proposed by Blomgren and Chan in 1998. We treat the diffusion matrixDimplicitly as the productD=LLT, and work with the elements ofLas variables, instead of working directly on the elements ofD. This ensures positive definiteness of the tensor during the regularization flow, which is essential when regularizing DTI. We perform numerical experiments on both synthetical data and 3D human brain DTI, and measure the quantitative behavior of the proposed model.

Ensemble-based data-assimilation methods have gained fast growth since the ensemble Kalman filter (EnKF) was introduced into the petroleum engineering. Many techniques have been developed to overcome the inherent disadvantages of the ensemble-based methods including the assumptions of linearity and Gaussianity, to make it more robust and reliable for practical reservoir applications. The current trend in petroleum reservoir history matching is towards taking into account more realistic reservoir models with complex geology. Geologic facies modeling plays an important role in the reservoir characterization as a way to reproduce important patterns of heterogeneity in petroleum reservoirs and to facilitate the modeling of petrophysical properties of reservoir rocks. Because the static data and general geological knowledge are almost always not sufficient to determine the distribution of geologic facies uniquely, it is advantageous to assimilate dynamic data to reduce the uncertainty. The history-matching problem of geologic facies involves large nonlinearity and non-Gaussianity. When the ensemble Kalman filter (EnKF) or related ensemble-based methods are used to assimilate data in a straightforward way, the updated model variables would lose the geological realism and be contaminated with noise. Therefore, it is necessary to develop some effective measures to adapt the ensemble-based dataassimilation methods to the facies problems. This thesis is motivated by this necessity and explores the ways to conditioning geologic facies to production data using ensemble-based methods. The focus is placed on the post-processing approach. By modifying the standard randomized maximum likelihood algorithm to accommodate non-Gaussian problems, we introduce a methodology that consists of a straightforward implementation of ensemble-based data assimilation in the first place and a sequential optimization procedure without iteration as a follow-up. In a similar manner, we develop another method for the post-processing of the updated reservoir models after data assimilation using ensemble-based methods. In the post-processing step, the objective function is composed of a weighted quadratic term measuring the distance to the posterior realizations, and a penalty term forcing model variables to take discrete values. A special emphasis is put on the investigation of the importance of the correlation information among the updated model variables introduced by the data which is usually ignored for the probability map approach. All of the proposed methodologies are evaluated via numerical experiments and demonstrated their utilities for improving the assimilation of data to geological facies models. In this thesis, we also investigate the application of discrete curvelet transform in the denoising problem of updated model variables by the direct use of ensemble-based data-assimilation methods. According to the numerical experiments, the results show that curvelets are useful for denoising in the problem concerned but lose data match unless the covariance is included.

We describe a graph parametrization of rational quadratic differentials with presence of a simple pole, whose critical trajectories form a network depending on parameters focusing on the network topological jumps. Obtained bifurcation diagrams are associated with the Stasheff polytopes., Comment: 14 pages, 8 figures

Let X be a standard determinantal scheme X \subset \PP^n of codimension c, i.e. a scheme defined by the maximal minors of a t \times (t+c-1) homogeneous polynomial matrix A. In this paper, we study the main features of its normal sheaf \shN_X. We prove that under some mild restrictions: (1) there exists a line bundle \shL on X \setminus Sing(X) such that \shN_X \otimes \shL is arithmetically Cohen-Macaulay and, even more, it is Ulrich whenever the entries of A are linear forms, (2) \shN_X is simple (hence, indecomposable) and, finally, (3) \shN_X is \mu-(semi)stable provided the entries of A are linear forms., Comment: This version makes a correction to Proposition 3.9 of previous versions on the arXiv, as well as to the published version in Crelle's journal, see Remark 3.18 for details

The multimode resource-constrained project scheduling problem (MRCPSP) has been confirmed to be an NP-hard problem. Particle swarm optimization (PSO) has been efficiently applied to the search for near optimal solutions to various NP-hard problems. MRCPSP involves solving two subproblems: mode assignment and activity priority determination. Hence, two PSOs are applied to each subproblem. A constriction PSO is proposed for the activity priority determination while a discrete PSO is employed for mode assignment. A least total resource usage (LTRU) heuristic and minimum slack (MSLK) heuristic ensure better initial solutions. To ensure a diverse initial collection of solutions and thereby enhancing the PSO efficiency, a best heuristic rate (HR) is suggested. Moreover, a new communication topology with random links is also introduced to prevent slow and premature convergence. To verify the performance of the approach, the MRCPSP benchmarks in PSPLIB were evaluated and the results compared to other state-of-the-art algorithms. The results demonstrate that the proposed algorithm outperforms other algorithms for the MRCPSP problems. Finally, a real-world man-day project scheduling problem (MDPSP)—a MRCPSP problem—was evaluated and the results demonstrate that MDPSP can be solved successfully.