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• 251.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Bounds on Hilbert Functions and Betti Numbers of Veronese Modules2014Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

The thesis is a collection of four papers dealing with Hilbert functions and Betti numbers.In the first paper, we study the h-vectors of reduced zero-dimensional schemes in $\mathbb{P}^{2}$ . In particular we deal with the problem of findingthe possible h-vectors for the union of two sets of points of given h-vectors. To answer to this problem, we give two kinds of bounds on theh-vectors and we provide an algorithm that calculates many possibleh-vectors.In the second paper, we prove a generalization of Green’s Hyper-plane Restriction Theorem to the case of finitely generated modulesover the polynomial ring, providing an upper bound for the Hilbertfunction of the general linear restriction of a module M in a degree dby the corresponding Hilbert function of a lexicographic module.In the third paper, we study the minimal free resolution of theVeronese modules, $S_{n,d,k}=\oplus_{i\geq 0} S_{k+id}$, where  $S = \mathbb{K}[x_1 , . . . , x_n ]$ by giving a formula for the Betti numbers in terms of the reduced homology of the squarefree divisor complex. We prove that $S_{n,d,k}$ is Cohen-Macaulay if and only if k < d, and that its minimal resolutionis linear when k > d(n − 1) − n. We prove combinatorially that the resolution of $S_{2,d,k}$ is pure. We provide a formula for the Hilbert seriesof the Veronese modules. As an application, we calculate the completeBetti diagrams of the Veronese rings $S_{3,4,0} , S_{3,5,0} and S_{4,3,0}$ .In the fourth paper, we apply the same combinatorial techniques inthe study of the properties of pinched Veronese rings, in particular weshow when this ring is Cohen-Macaulay. We also study the canonicalmodule of the Veronese modules.

• 252.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Cohen-Macaulay Property of pinched Veronese Rings and Canonical Modules of Veronese  ModulesManuskript (preprint) (Övrigt vetenskapligt)

In this paper, we study the Betti numbers of pinched Veronese rings, by means of the reduced homology of the squarefree divisor complex. In particular, we study the Cohen-Macaulay property of these rings. Moreover, in the last section we compute the canonical modules of the Veronese modules.

• 253.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Syzygies of Veronese modulesManuskript (preprint) (Övrigt vetenskapligt)
• 254. Greco, Ornella
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Syzygies of Veronese ModulesManuskript (preprint) (Övrigt vetenskapligt)
• 255.
Univ Calif Riverside, Dept Math, Riverside, CA 92521 USA..
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
Koszul Duality for Semidirect Products and Generalized Takiff Algebras2017Ingår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 20, nr 3, s. 675-694Artikel i tidskrift (Refereegranskat)

We obtain Koszul-type dualities for categories of graded modules over a graded associative algebra which can be realized as the semidirect product of a bialgebra coinciding with its degree zero part and a graded module algebra for the latter. In particular, this applies to graded representations of the universal enveloping algebra of the Takiff Lie algebra (or the truncated current algebra) and its (super)analogues, and also to semidirect products of quantum groups with braided symmetric and exterior module algebras in case the latter are flat deformations of classical ones.

• 256.
Univ Bielefeld, Fac Math, POB 100 131, D-33501 Bielefeld, Germany..
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
Finitary 2-categories associated with dual projection functors2017Ingår i: Communications in Contemporary Mathematics, ISSN 0219-1997, Vol. 19, nr 3, artikel-id 1650016Artikel i tidskrift (Refereegranskat)

We study finitary 2-categories associated to dual projection functors for finite-dimensional associative algebras. In the case of path algebras of admissible tree quivers (which includes all Dynkin quivers of type A), we show that the monoid generated by dual projection functors is the Hecke-Kiselman monoid of the underlying quiver and also obtain a presentation for the monoid of indecomposable subbimodules of the identity bimodule.

• 257.
Aalto Univ, Dept Math & Syst Anal, Aalto 00076, Finland..
Aalto Univ, Dept Math & Syst Anal, Aalto 00076, Finland.. Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik. Aalto Univ, Dept Math & Syst Anal, Aalto 00076, Finland..
Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes2019Ingår i: IEEE Transactions on Information Theory, ISSN 0018-9448, E-ISSN 1557-9654, Vol. 65, nr 10, s. 6089-6100, artikel-id 8700214Artikel i tidskrift (Refereegranskat)

Locally repairable codes (LRCs) have gained significant interest for the design of large distributed storage systems as they allow a small number of erased nodes to be recovered by accessing only a few others. Several works have thus been carried out to understand the optimal rate-distance tradeoff, but only recently the size of the alphabet has been taken into account. In this paper, a novel definition of locality is proposed to keep track of the precise number of nodes required for a local repair when the repair sets do not yield MDS codes. Then, a new alphabet-dependent bound is derived, which applies both to the new definition and the initial definition of locality. The new bound is based on consecutive residual codes and intrinsically uses the Griesmer bound. A special case of the bound yields both the extension of the Cadambe-Mazumdar bound and the Singleton-type bound for codes with locality (r, delta), implying that the new bound is at least as good as these bounds. Furthermore, an upper bound on the asymptotic rate-distance tradeoff of LRCs is derived, and yields the tightest known upper bound for large relative minimum distances. Achievability results are also provided by deriving the locality of the family of Simplex codes together with a few examples of optimal codes.

• 258.
KTH, Skolan för datavetenskap och kommunikation (CSC), Teoretisk datalogi, TCS.
Scania. KTH, Skolan för industriell teknik och management (ITM), Maskinkonstruktion (Inst.), Mekatronik. Systems Development DivisionScania AB, Södertälje, Sweden. KTH, Skolan för industriell teknik och management (ITM), Maskinkonstruktion (Inst.), Mekatronik.
Deductive Functional Verification of Safety-Critical Embedded C-Code: An Experience Report2017Ingår i: Critical Systems: Formal Methods and Automated Verification, Cham: Springer, 2017, s. 3-18Konferensbidrag (Refereegranskat)

This paper summarizes our experiences from an exercise in deductive verification of functional properties of automotive embedded Ccode in an industrial setting. We propose a formal requirements model that supports the way C-code requirements are currently written at Scania. We describe our work, for a safety-critical module of an embedded system, on formalizing its functional requirements and verifying its C-code implementation by means of VCC, an established tool for deductive verification. We describe the obstacles we encountered, and discuss the automation of the specification and annotation effort as a prerequisite for integrating this technology into the embedded software design process.

• 259.
KTH, Matemiska institutionen.
KTH, Matemiska institutionen.
The resultant on compact Riemann surfaces2009Ingår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 286, nr 1, s. 313-358Artikel i tidskrift (Refereegranskat)

We introduce a notion of resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact Riemann surface. As a particular application, the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the Schottky double of the domain.

• 260.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
Representations of Finite-Dimensional Algebras and Gabriel’s Theorem2019Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
• 261.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
Post's problem in recursion theory and its solution2019Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
• 262.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
Talteori och kryptering2019Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
• 263.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
Construction of Irreducible Polynomials over Finite Fields2014Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
• 264.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
On division algebras of dimension 2n admitting Cn2 as a subgroup of their automorphism group2016Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)
• 265.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
Origins of Integration2016Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
• 266.
LIM&BIO (EA3969), Université Paris 13, France.
Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik. Université Paris 13, France. Universiteá Lille, Villeneuve d'Ascq, France. Universiteá Lille, Villeneuve d'Ascq, France. Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.
Combining Compositionality and Pagerank for the Identification of Semantic Relations between Biomedical Words2012Ingår i: BioNLP: Proceedings of the 2012 Workshop on Biomedical Natural Language Processing, 2012, s. 109-117Konferensbidrag (Refereegranskat)

The acquisition of semantic resources and relations is an important task for several applications, such as query expansion, information retrieval and extraction, machine translation. However, their validity should also be computed and indicated, especially for automatic systems and applications. We exploit the compositionality based methods for the acquisition of synonymy relations and of indicators of these synonyms. We then apply pagerank-derived algorithm to the obtained semantic graph in order to filter out the acquired synonyms. Evaluation performed with two independent experts indicates that the quality of synonyms is systematically improved by 10 to 15% after their filtering.

• 267.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
Admissible Domain Representations of Convergence Spaces2005Rapport (Övrigt vetenskapligt)
• 268.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
Admissible Domain Representations of Topological Spaces2005Rapport (Övrigt vetenskapligt)
• 269.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
An enquiry concerning categories of effective continuous cpos2002Licentiatavhandling, monografi (Övrigt vetenskapligt)
• 270.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
Effective Domains and Admissible Domain Representations2005Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

This thesis consists of four papers in domain theory and a summary. The first two papers deal with the problem of defining effectivity for continuous cpos. The third and fourth paper present the new notion of an admissible domain representation, where a domain representation D of a space X is λ-admissible if, in principle, all other λ-based domain representations E of X can be reduced to X via a continuous function from E to D.

In Paper I we define a cartesian closed category of effective bifinite domains. We also investigate the method of inducing effectivity onto continuous cpos via projection pairs, resulting in a cartesian closed category of projections of effective bifinite domains.

In Paper II we introduce the notion of an almost algebraic basis for a continuous cpo, showing that there is a natural cartesian closed category of effective consistently complete continuous cpos with almost algebraic bases. We also generalise the notion of a complete set, used in Paper I to define the bifinite domains, and investigate what closure results that can be obtained.

In Paper III we consider admissible domain representations of topological spaces. We present a characterisation theorem of exactly when a topological space has a λ-admissible and κ-based domain representation. We also show that there is a natural cartesian closed category of countably based and countably admissible domain representations.

In Paper IV we consider admissible domain representations of convergence spaces, where a convergence space is a set X together with a convergence relation between nets on X and elements of X. We study in particular the new notion of weak κ-convergence spaces, which roughly means that the convergence relation satisfies a generalisation of the Kuratowski limit space axioms to cardinality κ. We show that the category of weak κ-convergence spaces is cartesian closed. We also show that the category of weak κ-convergence spaces that have a dense, λ-admissible, κ-continuous and α-based consistently complete domain representation is cartesian closed when α ≤ λ ≥ κ. As natural corollaries we obtain corresponding results for the associated category of weak convergence spaces.

• 271.
Department of Mathematics, Uppsala university.
Effective Domains and Admissible Domain Representations2005Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

This thesis consists of four papers in domain theory and a summary. The first two papers deal with the problem of defining effectivity for continuous cpos. The third and fourth paper present the new notion of an admissible domain representation, where a domain representation D of a space X is λ-admissible if, in principle, all other λ-based domain representations E of X can be reduced to X via a continuous function from E to D. In Paper I we define a cartesian closed category of effective bifinite domains. We also investigate the method of inducing effectivity onto continuous cpos via projection pairs, resulting in a cartesian closed category of projections of effective bifinite domains. In Paper II we introduce the notion of an almost algebraic basis for a continuous cpo, showing that there is a natural cartesian closed category of effective consistently complete continuous cpos with almost algebraic bases. We also generalise the notion of a complete set, used in Paper I to define the bifinite domains, and investigate what closure results that can be obtained. In Paper III we consider admissible domain representations of topological spaces. We present a characterisation theorem of exactly when a topological space has a λ-admissible and κ-based domain representation. We also show that there is a natural cartesian closed category of countably based and countably admissible domain representations. In Paper IV we consider admissible domain representations of convergence spaces, where a convergence space is a set X together with a convergence relation between nets on X and elements of X. We study in particular the new notion of weak κ-convergence spaces, which roughly means that the convergence relation satisfies a generalisation of the Kuratowski limit space axioms to cardinality κ. We show that the category of weak κ-convergence spaces is cartesian closed. We also show that the category of weak κ-convergence spaces that have a dense, λ-admissible, κ-continuous and α-based consistently complete domain representation is cartesian closed when α ≤ λ ≥ κ. As natural corollaries we obtain corresponding results for the associated category of weak convergence spaces.

• 272.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
Two Categories of Effective Continuous Cpos2005Rapport (Övrigt vetenskapligt)
• 273.
Matematiska institutionen, Uppsala universitet.
Matematiska institutionen, Uppsala universitet.
Cartesian closed categories of effective domains2001Konferensbidrag (Refereegranskat)
• 274.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
Cartesian closed categories ofeffective domains2002Ingår i: Proof and System-Reliability, s. 1-20Artikel i tidskrift (Refereegranskat)
• 275.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
Effective Cartesian Closed Categories of Domains2001Rapport (Övrigt vetenskapligt)
• 276.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
Two categories of effective continuous cpos2006Ingår i: Theoretical Computer Science, Vol. 365, nr 3, s. 216-236Artikel i tidskrift (Refereegranskat)
• 277.
Uppsala universitet, Humanistisk-samhällsvetenskapliga vetenskapsområdet, Historisk-filosofiska fakulteten, Filosofiska institutionen. Teoretisk filosofi.
A Study of König's Lemma2002Ingår i: Uppsala Prints and Preprints in Philosophy, nr 2, s. 61-Artikel, forskningsöversikt (Övrig (populärvetenskap, debatt, mm))
• 278.
Uppsala universitet, Humanistisk-samhällsvetenskapliga vetenskapsområdet, Historisk-filosofiska fakulteten, Filosofiska institutionen. Teoretisk filosofi.
Applied Logic: Logical Foundations of Logic Programming2004Övrigt (Övrigt vetenskapligt)
• 279.
Uppsala universitet, Humanistisk-samhällsvetenskapliga vetenskapsområdet, Historisk-filosofiska fakulteten, Filosofiska institutionen. Teoretisk filosofi.
New Definitions of the Recursive Functions2002Ingår i: Uppsala Prints and Preprints in Philosophy, nr 6, s. 45-Artikel, forskningsöversikt (Övrig (populärvetenskap, debatt, mm))

Four new definitions are given of the recursive functions. They are proven equivalent with Kleene's definition.

• 280.
Uppsala universitet, Humanistisk-samhällsvetenskapliga vetenskapsområdet, Historisk-filosofiska fakulteten, Filosofiska institutionen.
Kanger's Ideas on Non-Well-Founded Sets: Some Remarks2001Ingår i: Collected Papers of Stig Kanger with Essays on his Life and Work: Volume II, Kluwer Academic Publishers, Dordrecht, The Netherlands , 2001, s. 69-86Kapitel i bok, del av antologi (Övrigt vetenskapligt)
• 281.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
Vektorprodukten: historik och egenskaper2014Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
• 282.
KTH, Skolan för arkitektur och samhällsbyggnad (ABE), Filosofi och teknikhistoria, Filosofi.
Decomposition of multiple AGM contraction: possibility and impossibility results2014Ingår i: Logic journal of the IGPL (Print), ISSN 1367-0751, E-ISSN 1368-9894, Vol. 22, nr 4, s. 696-710Artikel i tidskrift (Refereegranskat)

Partial meet contraction, the standard operation of contraction in AGM theory, can straightforwardly be generalized to contractions with sets of sentences instead of single sentences as inputs (multiple contraction). The conditions under which multiple contraction can be reconstructed as the intersection of several contractions by single sentences are investigated. Although such reconstruction is possible in some special cases, the major result is that in the general case, full reconstruction is not possible. Therefore, multiple contraction adds to the expressive power of belief revision theory. It is concluded that this result highlights the importance of studying multiple belief change, and multiple conclusion logic in general.

• 283.
University of Leicester, UK.
Linnéuniversitetet, Fakulteten för teknik (FTK), Institutionen för matematik (MA).
Statistical and subjective interpretations of probability in quantum-like models of cognition and decision making2016Ingår i: Journal of mathematical psychology (Print), ISSN 0022-2496, E-ISSN 1096-0880, Vol. 74, s. 82-91Artikel i tidskrift (Refereegranskat)

The paper starts with an introduction to the basic mathematical model of classical probability (CP), i.e. the Kolmogorov (1933) measure-theoretic model. Its two basic interpretations are discussed: statistical and subjective. We then present the probabilistic structure of quantum mechanics (QM) and discuss the problem of interpretation of a quantum state and the corresponding probability given by Born’s rule. Applications of quantum probability (QP) to modeling of cognition and decision making (DM) suffer from the same interpretational problems as QM. Here the situation is even more complicated than in physics. We analyze advantages and disadvantages of the use of subjective and statistical interpretations of QP. The subjective approach to QP was formalized in the framework of Quantum Bayesianism (QBism) as the result of efforts from C. Fuchs and his collaborators. The statistical approach to QP was presented in a variety of interpretations of QM, both in nonrealistic interpretations, e.g., the Copenhagen interpretation (with the latest version due to A. Plotnitsky), and in realistic interpretations (e.g., the recent Växjö interpretation). At present, we cannot make a definite choice in favor of any of the interpretations. Thus, quantum-like DM confronts the same interpretational problem as quantum physics does.

• 284.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra, geometri och logik.
Contributions to Pointfree Topology and Apartness Spaces2011Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

The work in this thesis contains some contributions to constructive point-free topology and the theory of apartness spaces. The first two papers deal with constructive domain theory using formal topology. In Paper I we focus on the notion of a domain representation of a formal space as a way to introduce generalized points of the represented space, whereas we in Paper II give a constructive and point-free treatment of the domain theoretic approach to differential calculus. The last two papers are of a slightly different nature but still concern constructive topology. In paper III we consider a measure theoretic covering theorem from various constructive angles in both point-set and point-free topology. We prove a point-free version of the theorem. In Paper IV we deal with issues of impredicativity in the theory of apartness spaces. We introduce a notion of set-presented apartness relation which enables a predicative treatment of basic constructions of point-set apartness spaces.

• 285.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra, geometri och logik.
Local Scott Compactification2011Rapport (Övrigt vetenskapligt)
• 286.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra, geometri och logik.
Local Scott compactificationArtikel i tidskrift (Refereegranskat)

We show how to embed certain formal topologies in locally Scott formal topologies. We call this process local Scott compactification. Examples include localic completions of metric spaces. We also prove a lifting result for morphisms. The local Scott compactification of a space corresponds to a domain representation of the space and in this way yields a space containing partial elements. In the case of the real numbers we obtain the space of partial reals, consisting of intervals where the endpoints are lower and upper reals respectively. We show that partial reals that define compact overt subspaces of the real numbers correspond precisely to intervals with real endpoints. The lifting of morphisms give sharp extensions of continuous real valued functions in the sense of interval analysis. These results are also generalized to normed vector spaces.

• 287.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra, geometri och logik.
The Domain Theoretic Derivative in Formal Topology2011Rapport (Övrigt vetenskapligt)
• 288.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
2-Categories and Yoneda lemma2016Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
• 289.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Some remarks on biequidimensionality of topological spaces and Noetherian schemesManuskript (preprint) (Övrigt vetenskapligt)

There are many examples of the fact that dimension and codimension behave somewhat counterintuitively. In EGA it is stated that a topological space is equidimensional, equicodimensional and catenary if and only if every maximal chain of irreducible closed subsets has the same length. We construct examples that show that this is not even true for the spectrum of a Noetherian ring. This gives rise to two notions of biequidimensionality, and we show how these relate to the dimension formula and the existence of a codimension function.

• 290.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
The Cohen–Macaulay space of twisted cubicsManuskript (preprint) (Övrigt vetenskapligt)

In this work, we describe the Cohen-Macaulay space CM of twisted cubics parameterizing curves $C$ together with a finite map $i: C\rightarrow \mathbb{P}^3$ that is generically a closed immersion and such that $C$ has Hilbert polynomial p(t)=3t+1 with respect to $i$. We show that CM is irreducible, smooth and birational to one component of the Hilbert scheme of twisted cubics.

• 291.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
The space of Cohen–Macaulay curvesManuskript (preprint) (Övrigt vetenskapligt)

One can consider the Hilbert scheme as a natural compactification of the space of smooth projective curves with fixed Hilbert polynomial. Here we consider a different modular compactification, namely the functor CM parameterizing curves together with a finite map to $\mathbb{P}^{n}$ that is generically a closed immersion. We prove that CM is an algebraic space by contructing a scheme W and a representable, surjective and smooth map W -> CM. Moreover, we show that CM satisfies the valuative criterion for properness.

• 292.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
The space of Cohen-Macaulay curves2012Licentiatavhandling, monografi (Övrigt vetenskapligt)

In this thesis we discuss a moduli space of projective curves with a map to a given projective space. The functor CM parametrizes curves, that is, Cohen-Macaulay schemes of pure dimension 1, together with a finite map to the projective space that is an isomorphism onto its image away from a finite set of closed points.

We proof that CM is an algebraic space by contructing a scheme W and a representable, surjective and smooth map from W to CM.

• 293.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
The space of Cohen–Macaulay curves and related topics2014Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

The space of Cohen–Macaulay curves is a compactification of the space of curves that are embedded in a given projective space Pn. The idea is similar to that of the Hilbert scheme but instead of adding degenerated curves, one considers only curves without embedded or isolated points. However, the curves need not be embedded into the projective space. Instead, they come with a finite morphism to Pn that is generically a closed immersion. More precisely, the space CM of Cohen–Macaulay curves parameterizes flat families of pairs $(C,i)$ where $C$ is a curve without embedded or isolated points and $i: C\rightarrow \mathbb{P}^n$ is a finite morphism that is an isomorphism onto its image away from finitely many closed points and such that $C$ has Hilbert polynomial p(t) with respect to the map $i$.

In Paper A we show that the moduli functor CM is represented by a proper algebraic space. This is done by constructing a smooth, surjective cover $\pi: W\rightarrow CM$ and by verifying the valuative criterion for properness.

Paper B studies the moduli space CM in the particular case n = 3 and p(t) = 3t + 1, that is, the Cohen–Macaulay space of twisted cubics. We de- scribe the points of CM and show that they are in bijection with the points on the 12-dimensional component H0 of the Hilbert scheme of twisted cu- bics. Knowing the points of CM, we can then show that the moduli space is irreducible, smooth and has dimension 12.

Paper C concerns the notion of biequidimensionality of topological spaces and Noetherian schemes. In EGA it is claimed that a topological space X is equidimensional, equicodimensional and catenary if and only if all maximal chains of irreducible closed subsets in X have the same length. We construct examples of topological spaces and Noetherian schemes showing that the sec- ond property is strictly stronger. This gives rise to two different notions of biequidimensionality, and we show how they relate to the dimension formula and the existence of a codimension function.

• 294.
Mälardalens högskola, Akademin för utbildning, kultur och kommunikation. Umeå University.
A Rewriting Approach to Graph Invariants2009Ingår i: Generalized Lie Theory in Mathematics, Physics and Beyond / [ed] Sergei Silvestrov, Eugen Paal, Viktor Abramov, Alexander Stolin, Berlin, Heidelberg: Springer, 2009, s. 47-67Kapitel i bok, del av antologi (Refereegranskat)

Diagrammatic calculation is a powerful tool that gets near indispensable when one tries to manage some of the newer algebraic structures that have been popping up in the last couple of decades. Concretely, it generalises the underlying structure of expressions to being general graphs, where traditional algebraic notation only supports path- or treelike expressions. This paper demonstrates how to apply the author's Generic Diamond Lemma in diagrammatic calculations, by solving through elementary rewriting techniques the problem of classifying all multigraph invariants satisfying a linear contract—delete recursion. (As expected, this leads one to rediscover the Tutte polynomial, along with some more degenerate invariants.) In addition, a concept of “semigraph” is defined which formalises the concept of a graph-theoretical “gadget”.

• 295.
Mälardalens högskola, Akademin för utbildning, kultur och kommunikation. Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.
Critical Pairs in Network Rewriting2014Ingår i: IWC 2014: 3rd International Workshop on Confluence / [ed] Takahito Aoto & Delia Kesner, 2014, s. 9-13Konferensbidrag (Refereegranskat)

This extended abstract breifly introduces rewriting of networks (directed acyclic graphs with the extra structure needed to serve as expressions for PROducts and Permutations categories) and describes the critical pairs aspects of this theory. The author's interest in these comes from wanting to do equational reasoning in algebraic theories (such as Hopf algebras) that mix ordinary operations with co-operations; networks then serve as a formalism for expressions.

The main message is to point out two phenomena that arise in network rewriting. The first is that of non-convexity of rules, wherein the left hand side of a rule need not be syntactically similar to a symbol in any extension of the underlying signature. The second is one of critical pairs potentially arising where two redexes wrap around each other even when they do not intersect.

• 296.
Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.
Mathematical Formulae Beyond the Tree Paradigm2016Konferensbidrag (Övrigt vetenskapligt)

A restriction in the traditional language of mathematics is that every formula has an underlying tree structure, wherein an expression decomposes into subexpressions that each contribute one result to the whole. Some modern mathematics (especially in higher algebra and differential geometry) is however not so well catered for by this tree paradigm, and would be much better off if this restriction was lifted; it is not that the mathematics requires a generalised formula language, but it becomes a whole lot easier to state and do when you use one.

This poster shows the formula language of networks that arises when one relaxes the tree restriction to instead allow an underlying directed acyclic graph (DAG) structure. Evaluation of networks is non-obvious, but can be carried out in any PROP/symocat; conversely, evaluation of networks characterises PROPs in much the same way as evaluation of words characterises monoids. Networks are isomorphic to the abstract index interpretation of Einstein convention tensor expressions, and demonstrate that these can in fact be given a coordinate-free interpretation. Special cases of the network notation that have been invented independently include the Penrose graphical notation and quantum gate arrays.

• 297.
Mälardalens högskola, Akademin för utbildning, kultur och kommunikation.
Network Rewriting I: The Foundation2012Rapport (Övrigt vetenskapligt)

A theory is developed which uses "networks" (directed acyclic graphs with some extra structure) as a formalism for expressions in multilinear algebra. It is shown that this formalism is valid for arbitrary PROPs (short for 'PROducts and Permutations category'), and conversely that the PROP axioms are implicit in the concept of evaluating a network. Ordinary terms and operads constitute the special case that the graph underlying the network is a rooted tree. Furthermore a rewriting theory for networks is developed. Included in this is a subexpression concept for which is given both algebraic and effective graph-theoretical characterisations, a construction of reduction maps from rewriting systems, and an analysis of the obstructions to confluence that can occur. Several Diamond Lemmas for this rewriting theory are given. In addition there is much supporting material on various related subjects. In particular there is a "toolbox" for the construction of custom orders on the free PROP, so that an order can be tailored to suit a specific rewriting system. Other subjects treated are the abstract index notation in a general PROP context and the use of feedbacks (sometimes called traces) in PROPs.

• 298.
Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.
Network Rewriting II: Bi- and Hopf Algebras2015Ingår i: Leibniz International Proceedings in Informatics, LIPIcs, Volume 36 / [ed] Maribel Fernández, Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik , 2015, Vol. 36, s. 194-208Konferensbidrag (Refereegranskat)

Bialgebras and their specialisation Hopf algebras are algebraic structures that challenge traditional mathematical notation, in that they sport two core operations that defy the basic functional paradigm of taking zero or more operands as input and producing one result as output. On the other hand, these peculiarities do not prevent studying them using rewriting techniques, if one works within an appropriate network formalism rather than the traditional term formalism. This paper restates the traditional axioms as rewriting systems, demonstrating confluence in the case of bialgebras and finding the (infinite) completion in the case of Hopf algebras. A noteworthy minor problem solved along the way is that of constructing a quasi-order with respect to which the rules are compatible.

• 299.
Mälardalens högskola, Akademin för utbildning, kultur och kommunikation.
Network Rewriting II: Bi- and Hopf AlgebrasManuskript (preprint) (Övrigt vetenskapligt)

Bialgebras and their specialisation Hopf algebras are algebraic structures that challenge traditional mathematical notation, in that they sport two core operations that defy the basic functional paradigm of taking zero or more operands as input and producing one result as output. On the other hand, these peculiarities do not prevent studying them using rewriting techniques, if one works within an appropriate network formalism. This paper restates the traditional axioms as rewriting systems, demonstrating confluence in the case of bialgebras and finding the (infinite) completion in the case of Hopf algebras. A noteworthy minor problem solved along the way is that of constructing a quasi-order with respect to which the rules are compatible.

• 300.
Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.
Ordered and Combinatorial Structures for Higher-Dimensional Rewriting2016Ingår i: / [ed] Samuel Mimram, 2016Konferensbidrag (Refereegranskat)

In principle, rewriting is the logically pure activity of transforming abstract expressions according to fully formalised rules, but in practice there is a significant interplay between abstract rules, more-or-less concrete interpretations, and a variety of book-keeping devices that all need to fit together if the rewriting process is to yield results. This paper presents elementary realisations of book-keeping and other formalising devices that are useful in higher-dimensional rewriting, with a focus on the "2-dimensional" case (PROPs and other types of monoidal category). In particular, it explains how one may construct a variety of ordering relations on these object that are sensitive to differences in the underlying graph structure of the objects being rewritten. It also shows how the formal feedback operation can be used to handle nonconvex redexes, which is a phenomenon of higher-dimensional rewriting that lacks a counterpart in word or term rewriting.

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