This thesis consists of a summary and three articles. The thesis is devoted to the study of knots and surfaces with additional geometric structures compared to the classical smooth structure. In Paper I, real algebraic rational knots in real projective space are studied up to rigid isotopy and we show that two real rational algebraic knots of degree at most 5 are rigidly isotopic if, and only if, their degree and encomplexed writhe are equal. We also show that any smooth irreducible knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree at most 6. Furthermore, an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented. In Paper II, we construct an invariant of parametrized generic real algebraic surfaces in real projective space which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: an intersection of two real sheets, an intersection of two complex conjugate sheets or a Whitney umbrella. The Brown invariant was expressed through a self linking number of the self intersection by Kirby and Melvin. We extend their definition of this self linking number to the case of parametrized generic real algebraic surfaces. In Paper III, we give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in the product of a punctured Riemann surface with the real line. As an application we show that for any nonzero homology class h, and for any integer k there exist k Legendrian knots all representing h which are pairwise smoothly isotopic through a formal Legendrian isotopy but which lie in mutually distinct Legendrian isotopy classes.

I en stor och komplex organisation kan det vara svårt att skapa sig en helhetsbild och se sitt arbete i relation till organisationens övergripande syfte. En sådan helhetsbild kan skapas genom att kartlägga och dokumentera aktivitetsflödet i organisationen. Processutveckling tillhandahåller verktyg för att eliminera dubbelarbete och spill och skapa bättre värde för organisationens kunder. Som resultat av detta förenklas arbetet, kostnader kan lättare reduceras och ledtider kan minskas. Allt detta sammantaget leder till ökad konkurrenskraft. Syftet med detta examensarbete är att införa grunderna för processutveckling på Siemens GPMT. Vårt mål är att introducera och etablera ett processinriktat synsätt samt skapa verktyg för att avdelningen i framtiden själv skall kunna utveckla sin verksamhet. Rapporten inleds med en presentation av företaget och den problembakgrund som har föranlett examensarbetet. Centrala teorier ochen allmänt applicerbar metodik för processutveckling beskrivs ingående. Slutligen presenteras de resultat vi kommit fram till under arbetets gång och en avslutande diskussion. Vi har skapat beskrivningar av arbetet samt underlag för mätning och uppföljning av processernas prestanda. Dessutom har vi tagit fram en ledningsstruktur för fortsatt arbete med processutveckling. Väldokumenterade processer och standardiserade arbetsrutiner är ingen garanti för goda resultat. De anger endast den befintliga kunskapsnivån som finns inom organisationen. Deras fulla potential kan endast utnyttjas om personalen verkligen tror på deras värde och arbetar för att utveckla och driva verksamheten framåt. Vi är övertygade om att GPMT, genom att ta till sig de resonemang som vi presenterat under vårt arbete och arbetar aktivt med de metoder som vi beskrivit, kommer att utvecklas till att bli en förebild för alla andra avdelningar på Siemens. / In a large and complex organization, it is often difficult to obtain acomplete picture and be able to see individual contributions to the overall objectives of the organization. A wide perspective can be created through mapping and documentation of the work-flow within the organization. Process management is a method containing tools that facilitates elimination of overlapping activities and waste, which leads to greater value for the customers. Through this method, work is simplified, costs can be reduced and lead-times shortened. The end result is an improved market position. The purpose of this thesis work is to set up the fundamentals of process management at Siemens GPMT. Our goal is to introduce and establish a process oriented perspective and provide tools for the department to be able to improve their processes. The report starts with a presentation of the company and the problembackground that has led to this thesis. Essential theories and methods for process management are thoroughly described. The results we have found under the project are presented along with a final discussion and conclusions. We have created work descriptions along with tools for measuring and evaluating the performance of the processes. We have also created a structure for further work with process management. Well documented processes and work standards are no guarantee for good results. They only present the current level of knowledge within the organization. Their full potential can only be utilized if the personnel really believe in their value and strive to continuously improve the business. We are firmly convinced that GPMT, by adopting our recommendations and actively working with the methods we have described, will develop to be a role model for all other departments at Siemens.

Syftet med detta examensarbete är att undersöka om och hur elever uppfattar begreppen omkrets och area. Då vi båda studerar till matematiklärare anser vi att detta arbete kan stärka oss i vår lärarroll samt utveckla oss inom arbetsområdet geometri. Vi började med att göra en diagnos med ca 40 elever. Genom att djupintervjua tre av eleverna fick vi en bättre förståelse för varför eleverna svarade som de gjorde på diagnosen. Genom att göra en läromedelsgranskning av de aktuella läroböckerna ville vi se om dessa kan spela roll när det gäller elevernas kunskap inom detta område. Resultatet visar att de flesta eleverna klarade att räkna uppgifter inom området omkrets eller area och använda sig av dessa begrepp när uppgiften var ”given”, på så sätt att eleverna får veta vad som ska räknas ut (area eller omkrets). Problemen dyker upp när eleverna får svårare uppgifter som är konstruerade på ett sådant sätt att de inte får en given metod att lösa uppgiften, det vill säga när det är oklart hur eleverna ska angripa uppgiften.

<p>I en stor och komplex organisation kan det vara svårt att skapa sig en helhetsbild och se sitt arbete i relation till organisationens övergripande syfte. En sådan helhetsbild kan skapas genom att kartlägga och dokumentera aktivitetsflödet i organisationen. Processutveckling tillhandahåller verktyg för att eliminera dubbelarbete och spill och skapa bättre värde för organisationens kunder. Som resultat av detta förenklas arbetet, kostnader kan lättare reduceras och ledtider kan minskas. Allt detta sammantaget leder till ökad konkurrenskraft.</p><p>Syftet med detta examensarbete är att införa grunderna för processutveckling på Siemens GPMT. Vårt mål är att introducera och etablera ett processinriktat synsätt samt skapa verktyg för att avdelningen i framtiden själv skall kunna utveckla sin verksamhet.</p><p>Rapporten inleds med en presentation av företaget och den problembakgrund som har föranlett examensarbetet. Centrala teorier ochen allmänt applicerbar metodik för processutveckling beskrivs ingående. Slutligen presenteras de resultat vi kommit fram till under arbetets gång och en avslutande diskussion.</p><p>Vi har skapat beskrivningar av arbetet samt underlag för mätning och uppföljning av processernas prestanda. Dessutom har vi tagit fram en ledningsstruktur för fortsatt arbete med processutveckling.</p><p>Väldokumenterade processer och standardiserade arbetsrutiner är ingen garanti för goda resultat. De anger endast den befintliga kunskapsnivån som finns inom organisationen. Deras fulla potential kan endast utnyttjas om personalen verkligen tror på deras värde och arbetar för att utveckla och driva verksamheten framåt.</p><p>Vi är övertygade om att GPMT, genom att ta till sig de resonemang som vi presenterat under vårt arbete och arbetar aktivt med de metoder som vi beskrivit, kommer att utvecklas till att bli en förebild för alla andra avdelningar på Siemens.</p> / <p>In a large and complex organization, it is often difficult to obtain acomplete picture and be able to see individual contributions to the overall objectives of the organization.</p><p>A wide perspective can be created through mapping and documentation of the work-flow within the organization. Process management is a method containing tools that facilitates elimination of overlapping activities and waste, which leads to greater value for the customers. Through this method, work is simplified, costs can be reduced and lead-times shortened. The end result is an improved market position.</p><p>The purpose of this thesis work is to set up the fundamentals of process management at Siemens GPMT. Our goal is to introduce and establish a process oriented perspective and provide tools for the department to be able to improve their processes.</p><p>The report starts with a presentation of the company and the problembackground that has led to this thesis. Essential theories and methods for process management are thoroughly described. The results we have found under the project are presented along with a final discussion and conclusions.</p><p>We have created work descriptions along with tools for measuring and evaluating the performance of the processes. We have also created a structure for further work with process management.</p><p>Well documented processes and work standards are no guarantee for good results. They only present the current level of knowledge within the organization. Their full potential can only be utilized if the personnel really believe in their value and strive to continuously improve the business.</p><p>We are firmly convinced that GPMT, by adopting our recommendations and actively working with the methods we have described, will develop to be a role model for all other departments at Siemens.</p>

<p>Syftet med detta examensarbete är att undersöka om och hur elever uppfattar begreppen omkrets</p><p>och area. Då vi båda studerar till matematiklärare anser vi att detta arbete kan stärka oss i vår</p><p>lärarroll samt utveckla oss inom arbetsområdet geometri.</p><p>Vi började med att göra en diagnos med ca 40 elever. Genom att djupintervjua tre av eleverna</p><p>fick vi en bättre förståelse för varför eleverna svarade som de gjorde på diagnosen.</p><p>Genom att göra en läromedelsgranskning av de aktuella läroböckerna ville vi se om dessa kan</p><p>spela roll när det gäller elevernas kunskap inom detta område.</p><p>Resultatet visar att de flesta eleverna klarade att räkna uppgifter inom området omkrets eller</p><p>area och använda sig av dessa begrepp när uppgiften var ”given”, på så sätt att eleverna får veta</p><p>vad som ska räknas ut (area eller omkrets). Problemen dyker upp när eleverna får svårare</p><p>uppgifter som är konstruerade på ett sådant sätt att de inte får en given metod att lösa uppgiften,</p><p>det vill säga när det är oklart hur eleverna ska angripa uppgiften.</p>

Thesis (M.S.)--Missouri State University, 2008. / "August 2008." Includes bibliographical references (leaf 37). Also available online.

<p>We study the power series ring R= K[[x₁,x₂,x₃,...]]on countably infinitely many variables, over a field K, and two particular K-subalgebras of it: the ring S, which is isomorphic to an inverse limit of the polynomial rings in finitely many variables over K, and the ring R', which is the largest graded subalgebra of R.</p><p>Of particular interest are the homogeneous, finitely generated ideals in R', among them the <i>generic ideals</i>. The definition of S as an inverse limit yields a set of <i>truncation homomorphisms</i> from S to K[x₁,...,x_n] which restrict to R'. We have that the truncation of a generic I in R' is a generic ideal in K[x₁,...,x_n]. It is shown in <b>Initial ideals of Truncated Homogeneous Ideals</b> that the initial ideal of such an ideal converge to the initial ideal of the corresponding ideal in R'. This initial ideal need no longer be finitely generated, but it is always <i>locally finitely generated</i>: this is proved in <b>Gröbner Bases in R'</b>. We show in <b>Reverse lexicographic initial ideals of generic ideals are finitely generated</b> that the initial ideal of a generic ideal in R' is finitely generated. This contrast to the lexicographic term order.</p><p> If I in R' is a homogeneous, locally finitely generated ideal, and if we write the Hilbert series of the truncated algebras K[x₁,...,x_n] module the truncation of I as q_n(t)/(1-t)ⁿ, then we show in <b>Generalized Hilbert Numerators </b>that the q_n's converge to a power series in t which we call the <i>generalized Hilbert numerator</i> of the algebra R'/I.</p><p>In <b>Gröbner bases for non-homogeneous ideals in R'</b> we show that the calculations of Gröbner bases and initial ideals in R' can be done also for some non-homogeneous ideals, namely those which have an <i>associated homogeneous ideal</i> which is locally finitely generated.</p><p>The fact that S is an inverse limit of polynomial rings, which are naturally endowed with the discrete topology, provides S with a topology which makes it into a complete Hausdorff topological ring. The ring R', with the subspace topology, is dense in R, and the latter ring is the Cauchy completion of the former. In <b>Topological properties of R'</b> we show that with respect to this topology, locally finitely generated ideals in R'are <i>closed</i>.</p>

Thesis (Ph. D.)--University of Oregon, 2006. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 83-84). Also available for download via the World Wide Web; free to University of Oregon users.

The topic of this thesis is the study of moduli spaces of zero-dimensional geometricobjects. The thesis consists of three articles each focusing on a particular moduli space.The first article concerns the Hilbert scheme Hilb(X). This moduli space parametrizesclosed subschemes of a fixed ambient scheme X. It has been known implicitly for sometime that the Hilbert scheme does not behave well when the scheme X is not separated.The article shows that the separation hypothesis is necessary in the sense thatthe component Hilb1(X) of Hilb(X) parametrizing subschemes of dimension zero andlength 1 does not exist if X is not separated.Article number two deals with the Chow scheme Chow 0,n(X) parametrizing zerodimensionaleffective cycles of length n on the given scheme X. There is a relatedconstruction, the Symmetric product Symn(X), defined as the quotient of the n-foldproduct X ×. . .×X of X by the natural action of the symmetric group Sn permutingthe factors. There is a canonical map Symn(X) " Chow0,n(X) that, set-theoretically,maps a tuple (x1, . . . , xn) to the cycle!nk=1 xk. In many cases this canonical map is anisomorphism. We explore in this paper some examples where it is not an isomorphism.This will also lead to some results concerning the question whether the symmetricproduct commutes with base change.The third article is related to the Fulton-MacPherson compactification of the configurationspace of points. Here we begin by considering the configuration space F(X, n)parametrizing n-tuples of distinct ordered points on a smooth scheme X. The schemeF(X, n) has a compactification X[n] which is obtained from the product Xn by a sequenceof blowups. Thus X[n] is itself not defined as a moduli space, but the pointson the boundary of X[n] may be interpreted as geometric objects called stable degenerations.It is then natural to ask if X[n] can be defined as a moduli space of stabledegenerations instead of as a blowup. In the third article we begin work towards ananswer to this question in the case where X = P2. We define a very general modulistack Xpv2 parametrizing projective schemes whose structure sheaf has vanishing secondcohomology. We then use Artin’s criteria to show that this stack is algebraic. Onemay define a stack SDX,n of stable degenerations of X and the goal is then to provealgebraicity of the stack SDX,n by using Xpv2. / QC 20100729

We define a moduli functor parametrizing finite maps from a projective (locally) Cohen-Macaulay curve to a fixed projective space. The definition of the functor includes a number of technical conditions, but the most important is that the map is almost everywhere an isomorphism onto its image. The motivation for this definition comes from trying to interpolate between the Hilbert scheme and the Kontsevich mapping space. The main result is that our functor is represented by a proper algebraic space. As applications we obtain a new proof of the existence of Macaulayfications for varieties, and secondly, interesting compactifications of the spaces of smooth curves in projective space. We illustrate this in the case of rational quartics, where the resulting space appears easier than the Hilbert scheme. / QC 20101022