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  • 201.
    Bosk, Daniel
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    En formalisering av matematiken i svensk gymnasieundervisning2011Självständigt arbete på avancerad nivå (yrkesexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)
    Abstract [sv]

    Denna studie undersöker hur formell matematik kan undervisas i den nya svenska gymnasieskolan med dess nya ämnesplan för matematik. I studien undersöks hur ett undervisningsmaterial i formell matematik motsvarande det inledandeinnehållet i kursen Matematik 1c kan se ut och huruvida denna matematik kan undervisas med gymnasieelever.

    Undersökningen genomfördes med elever från det naturvetenskapliga programmets andra årskurs. Majoriteten av dessa elever läste då kursen Matematik D. Eleverna beskrev sig själva som ej motiverade i matematik.

    Resultatet visar att innehållet i ämnesplanen kan presenteras med formell matematik. Detta både med avseende ämnesplanens krav på innehåll och atteleverna kan förstå innehållet. Ämnesplanen kräver dessutom att denna typ av matematik tas upp som en del av innehållet i kursen Matematik 1c.

    Resultatet visar också att eleverna är öppna för och vill ha mer formell matematik i undervisningen. De tyckte att det kändes ovant eftersom att de aldrig tidigare stött på denna typ av matematik, men vissa elever fann formell matematik som enklare än matematiken som normalt presenteras på lektionerna.

    Studien finner ingen anledning till att skjuta upp mötet med formell matematik till universitetsnivå. Elevernas engagemang för bevis och tillgodogörandet av innehållet talar också för att formell matematik kan introduceras i gymnasiekurserna. Studiens slutsats är således att nya gymnasieskolans kurs Matematik 1c kan formaliseras och öppna för en förnyad matematikundervisning.

  • 202. Boucksom, Sebastien
    et al.
    Favre, Charles
    Jonsson, Mattias
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Differentiability Of Volumes Of Divisors And A Problem Of Teissier2009Ingår i: Journal of Algebraic Geometry, ISSN 1056-3911, E-ISSN 1534-7486, Vol. 18, nr 2, s. 279-308Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We give an algebraic construction of the positive intersection products of pseudo-effective classes and use them to prove that the volume function on the Neron-Severi space of a projective variety is C-1-differentiable, expressing its differential as a positive intersection product. We also relate the differential to the restricted volumes. We then apply our differentiability result to prove an algebro-geometric version of the Diskant inequality in convex geometry, allowing us to characterize the equality case of the Khovanskii-Teissier inequalities for nef and big classes.

  • 203. Boucksom, Sebastien
    et al.
    Favre, Charles
    Jonsson, Mattias
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Valuations and plurisubharmonic singularities2008Ingår i: Publications of the Research Institute for Mathematical Sciences, ISSN 0034-5318, E-ISSN 1663-4926, Vol. 44, nr 2, s. 449-494Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We extend to higher dimensions some of the valuative analysis of singularities of plurisubharmonic (psh) functions developed by the first two authors. Following Kontsevich and Soibelman we describe the geometry of the space V of all normalized valuations on C[x(1), ..., x(n)] centered at the origin. It is a union of simplices naturally endowed with an affine structure. Using relative positivity properties of divisors living on modifications of C-n above the origin, we define formal psh functions on V, designed to be analogues of the usual psh functions. For bounded formal psh functions on V, we define a mixed Monge-Ampere operator which reflects the intersection theory of divisors above the origin of C-n. This operator associates to any (n - 1)-tuple of formal psh functions a positive measure of finite mass on V. Next, we show that the collection of Lelong numbers of a given germ a of a psh function at all infinitely near points induces a formal psh function u on V. When phi is a psh Holder weight in the sense of Demailly, the generalized Lelong number v(phi)(u) equals the integral of u against the Monge-Ampere measure of phi. In particular, any generalized Lelong number is an average of valuations. We also show how to compute the multiplier ideal of u and the relative type of u with respect to phi in the sense of Rashkovskii, in terms of u and phi.

  • 204. Bousquet-Melou, Mireille
    et al.
    Linusson, Svante
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Nevo, Eran
    On the independence complex of square grids2008Ingår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 27, nr 4, s. 423-450Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The enumeration of independent sets of regular graphs is of interest in statistical mechanics, as it corresponds to the solution of hard-particle models. In 2004, it was conjectured by Fendley et al., that for some rectangular grids, with toric boundary conditions, the alternating number of independent sets is extremely simple. More precisely, under a coprimality condition on the sides of the rectangle, the number of independent sets of even and odd cardinality always differ by 1. In physics terms, this means looking at the hard-particle model on these grids at activity -1. This conjecture was recently proved by Jonsson. Here we produce other families of grid graphs, with open or cylindric boundary conditions, for which similar properties hold without any size restriction: the number of independent sets of even and odd cardinality always differ by 0, +/- 1, or, in the cylindric case, by some power of 2. We show that these results reflect a stronger property of the independence complexes of our graphs. We determine the homotopy type of these complexes using Forman's discrete Morse theory. We find that these complexes are either contractible, or homotopic to a sphere, or, in the cylindric case, to a wedge of spheres. Finally, we use our enumerative results to determine the spectra of certain transfer matrices describing the hard-particle model on our graphs at activity -1. These results parallel certain conjectures of Fendley et al., proved by Jonsson in the toric case.

  • 205. Brasco, L.
    et al.
    Lindgren, Erik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case2017Ingår i: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 304, s. 300-354Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We prove that for p≥2, solutions of equations modeled by the fractional p-Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in Wloc 1,p and their gradients are in a fractional Sobolev space as well. The relevant estimates are stable as the fractional order of differentiation s reaches 1. © 2016 Elsevier Inc.

  • 206. Brasco, L.
    et al.
    Lindgren, Erik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Parini, E.
    The fractional Cheeger problem2014Ingår i: Interfaces and free boundaries (Print), ISSN 1463-9963, E-ISSN 1463-9971, Vol. 16, nr 3, s. 419-458Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Given an open and bounded set Omega subset of R-N, we consider the problem of minimizing the ratio between the s-perimeter and the N-dimensional Lebesgue measure among subsets of Omega. This is the non-local version of the well-known Cheeger problem. We prove various properties of optimal sets for this problem, as well as some equivalent formulations. In addition, the limiting behaviour of some nonlinear and non-local eigenvalue problems is investigated, in relation with this optimization problem. The presentation is as self-contained as possible.

  • 207. Breuer, Jonathan
    et al.
    Duits, Maurice
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients2017Ingår i: Journal of The American Mathematical Society, ISSN 0894-0347, E-ISSN 1088-6834, Vol. 30, nr 1, s. 27-66Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study fluctuations of linear statistics corresponding to smooth functions for certain biorthogonal ensembles. We study those biorthogonal ensembles for which the underlying biorthogonal family satisfies a finite term recurrence and describe the asymptotic fluctuations using right limits of the recurrence matrix. As a consequence, we show that whenever the right limit is a Laurent matrix, a central limit theorem holds. We will also discuss the implications for orthogonal polynomial ensembles. In particular, we obtain a central limit theorem for the orthogonal polynomial ensemble associated with any measure belonging to the Nevai class of an interval. Our results also extend previous results on unitary ensembles in the one-cut case. Finally, we will illustrate our results by deriving central limit theorems for the Hahn ensemble for lozenge tilings of a hexagon and for the Hermitian two matrix model.

  • 208. Breuer, Jonathan
    et al.
    Duits, Maurice
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Nonintersecting paths with a staircase initial condition2012Ingår i: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 17, s. 1-24Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider an ensemble of N discrete nonintersecting paths starting from equidistant points and ending at consecutive integers. Our first result is an explicit formula for the correlation kernel that allows us to analyze the process as N -> infinity. In that limit we obtain a new general class of kernels describing the local correlations close to the equidistant starting points. As the distance between the starting points goes to infinity, the correlation kernel converges to that of a single random walker. As the distance to the starting line increases, however, the local correlations converge to the sine kernel. Thus, this class interpolates between the sine kernel and an ensemble of independent particles. We also compute the scaled simultaneous limit, with both the distance between particles and the distance to the starting line going to infinity, and obtain a process with number variance saturation, previously studied by Johansson.

  • 209. Breuer, Jonathan
    et al.
    Duits, Maurice
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Universality of Mesoscopic Fluctuations for Orthogonal Polynomial Ensembles2016Ingår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 342, nr 2, s. 491-531Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We prove that the fluctuations of mesoscopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical. Our main tool is an analysis of the Green's function for the associated Jacobi matrices. As a particular consequence we obtain a central limit theorem for the modified Jacobi Unitary Ensembles on all mesoscopic scales.

  • 210.
    Brocke, Ekaterina
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsvetenskap och beräkningsteknik (CST).
    Bhalla, Upinder S.
    Djurfeldt, Mikael
    Hällgren Kotaleski, Jeanette
    KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsvetenskap och beräkningsteknik (CST).
    Hanke, Michael
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Efficient Integration of Coupled Electrical-Chemical Systems in Multiscale Neuronal Simulations2016Ingår i: Frontiers in Computational Neuroscience, ISSN 1662-5188, E-ISSN 1662-5188, Vol. 10, artikel-id 97Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Multiscale modeling and simulations in neuroscience is gaining scientific attention due to its growing importance and unexplored capabilities. For instance, it can help to acquire better understanding of biological phenomena that have important features at multiple scales of time and space. This includes synaptic plasticity, memory formation and modulation, homeostasis. There are several ways to organize multiscale simulations depending on the scientific problem and the system to be modeled. One of the possibilities is to simulate different components of a multiscale system simultaneously and exchange data when required. The latter may become a challenging task for several reasons. First, the components of a multiscale system usually span different spatial and temporal scales, such that rigorous analysis of possible coupling solutions is required. Then, the components can be defined by different mathematical formalisms. For certain classes of problems a number of coupling mechanisms have been proposed and successfully used. However, a strict mathematical theory is missing in many cases. Recent work in the field has not so far investigated artifacts that may arise during coupled integration of different approximation methods. Moreover, in neuroscience, the coupling of widely used numerical fixed step size solvers may lead to unexpected inefficiency. In this paper we address the question of possible numerical artifacts that can arise during the integration of a coupled system. We develop an efficient strategy to couple the components comprising a multiscale test problem in neuroscience. We introduce an efficient coupling method based on the second-order backward differentiation formula (BDF2) numerical approximation. The method uses an adaptive step size integration with an error estimation proposed by Skelboe (2000). The method shows a significant advantage over conventional fixed step size solvers used in neuroscience for similar problems. We explore different coupling strategies that define the organization of computations between system components. We study the importance of an appropriate approximation of exchanged variables during the simulation. The analysis shows a substantial impact of these aspects on the solution accuracy in the application to our multiscale neuroscientific test problem. We believe that the ideas presented in the paper may essentially contribute to the development of a robust and efficient framework for multiscale brain modeling and simulations in neuroscience.

  • 211. Browning, T. D.
    et al.
    Matthiesen, Lilian
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Norm forms for arbitrary number fields as products of linear polynomials2017Ingår i: Annales Scientifiques de l'Ecole Normale Supérieure, ISSN 0012-9593, E-ISSN 1873-2151, Vol. 50, nr 6, s. 1383-1446Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Given a number field K=Q and a polynomial P ∈ Q[t], all of whose roots are in Q, let X be the variety defined by the equation NK(x) D P(t). Combining additive combinatorics with descent we show that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth and projective model of X.

  • 212. Bryan, Jim
    et al.
    Pandharipande, Rahul
    Faber, Carel
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Okounkov, A.
    The local Gromov-Witten theory of curves2008Ingår i: Journal of The American Mathematical Society, ISSN 0894-0347, E-ISSN 1088-6834, Vol. 21, nr 1, s. 101-136Artikel i tidskrift (Refereegranskat)
  • 213. Brzezinski, J.
    et al.
    Holsztynski, W.
    Kurlberg, Pär
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    On the congruence ax plus by 1 modulo xy2005Ingår i: Experimental Mathematics, ISSN 1058-6458, E-ISSN 1944-950X, Vol. 14, nr 4, s. 391-401Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We give bounds on the number of solutions to the Diophantine equation (X + 1/x)(Y + 1/y) = n as n tends to infinity. These bounds are related to the number of solutions to congruences of the form ax + by equivalent to 1 modulo xy.

  • 214.
    Bränden, Petter
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Moci, Luca
    The multivariate arithmetic Tutte polynomial2014Ingår i: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 366, nr 10, s. 5523-5540Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We introduce an arithmetic version of the multivariate Tutte polynomial and a quasi-polynomial that interpolates between the two. A generalized Fortuin-Kasteleyn representation with applications to arithmetic colorings and flows is obtained. We give a new and more general proof of the positivity of the coefficients of the arithmetic Tutte polynomial and (in the representable case) a geometrical interpretation of them.

  • 215.
    Bränden, Petter
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Wagner, David G.
    A converse to the Grace-Walsh-Szego theorem2009Ingår i: Mathematical proceedings of the Cambridge Philosophical Society (Print), ISSN 0305-0041, E-ISSN 1469-8064, Vol. 147, s. 447-453Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We prove that the symmetrizer of a permutation group preserves stability if and only if the group is orbit homogeneous. A consequence is that the hypothesis of permutation invariance in the Grace-Walsh-Szego Coincidence Theorem cannot be relaxed. In the process we obtain a new characterization of the Grace-like polynomials, introduced by D. Ruelle, and prove that the class of such polynomials can be endowed with a natural multiplication.

  • 216.
    Brändén, Petter
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Hyperbolicity cones of elementary symmetric polynomials are spectrahedral2014Ingår i: Optimization Letters, ISSN 1862-4472, E-ISSN 1862-4480, Vol. 8, nr 5, s. 1773-1782Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We prove that the hyperbolicity cones of elementary symmetric polynomials are spectrahedral, i.e., they are slices of the cone of positive semidefinite matrices. The proof uses the matrix-tree theorem, an idea already present in Choe et al.

  • 217.
    Brändén, Petter
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Solutions to two problems on permanents2012Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 436, nr 1, s. 53-58Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this note we settle two open problems in the theory of permanents by using recent results from other areas of mathematics. Both problems were recently discussed in Bapat's survey [2]. Bapat conjectured that certain quotients of permanents, which generalize symmetric function means, are concave. We prove this conjecture by using concavity properties of hyperbolic polynomials. Motivated by problems on random point processes, Shirai and Takahashi raised the problem: Determine all real numbers a for which the alpha-permanent (or alpha-determinant) is nonnegative for all positive semidefinite matrices. We give a complete solution to this problem by using recent results of Scott and Sokal on completely monotone functions. It turns out that the conjectured answer to the problem is false.

  • 218.
    Brändén, Petter
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Chasse, Matthew
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Infinite log-concavity for polynomial pólya frequency sequences2015Ingår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 143, nr 12, s. 5147-5158Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    McNamara and Sagan conjectured that if a0, a1, a2, . . . is a Pólya frequency (PF) sequence, then so is (formula presented), . . .. We prove this conjecture for a natural class of PF-sequences which are interpolated by polynomials. In particular, this proves that the columns of Pascal’s triangle are infinitely log-concave, as conjectured by McNamara and Sagan. We also give counterexamples to the first mentioned conjecture. Our methods provide families of nonlinear operators that preserve the property of having only real and nonpositive zeros.

  • 219.
    Brändén, Petter
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Haglund, James
    Visontai, Mirko
    Wagner, David G.
    Proof of the Monotone Column Permanent Conjecture2011Ingår i: Notions of Positivity and the Geometry of Polynomials / [ed] Petter Brändén, Mikael Passare, Mihai Putinar, Birkhäuser Verlag, 2011, 1, s. 63-78Kapitel i bok, del av antologi (Refereegranskat)
  • 220.
    Brändén, Petter
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Jonasson, Johan
    Negative Dependence in Sampling2012Ingår i: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 39, nr 4, s. 830-838Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    . The strong Rayleigh property is a new and robust negative dependence property that implies negative association; in fact it implies conditional negative association closed under external fields (CNA+). Suppose that and are two families of 0-1 random variables that satisfy the strong Rayleigh property and let . We show that {Zi} conditioned on is also strongly Rayleigh; this turns out to be an easy consequence of the results on preservation of stability of polynomials of Borcea & Branden (Invent. Math., 177, 2009, 521569). This entails that a number of important pps sampling algorithms, including Sampford sampling and Pareto sampling, are CNA+. As a consequence, statistics based on such samples automatically satisfy a version of the Central Limit Theorem for triangular arrays.

  • 221.
    Brändén, Petter
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Leander, M.
    Visontai, Mirkó
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Multivariate Eulerian Polynomials and Exclusion Processes2016Ingår i: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 25, nr 4, s. 486-499Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We give a new combinatorial interpretation of the stationary distribution of the (partially) asymmetric exclusion process on a finite number of sites in terms of decorated alternative trees and coloured permutations. The corresponding expressions of the multivariate partition functions are then related to multivariate generalisations of Eulerian polynomials for coloured permutations considered recently by N. Williams and the third author, and others. We also discuss stability and negative dependence properties satisfied by the partition functions.

  • 222.
    Brändén, Petter
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Moci, L.
    The multivariate arithmetic Tutte polynomial2012Ingår i: Discrete Mathematics & Theoretical Computer Science, ISSN 1462-7264, E-ISSN 1365-8050, s. 661-672Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We introduce an arithmetic version of the multivariate Tutte polynomial recently studied by Sokal, and a quasi-polynomial that interpolates between the two. We provide a generalized Fortuin-Kasteleyn representation for representable arithmetic matroids, with applications to arithmetic colorings and flows. We give a new proof of the positivity of the coefficients of the arithmetic Tutte polynomial in the more general framework of pseudo-arithmetic matroids. In the case of a representable arithmetic matroid, we provide a geometric interpretation of the coefficients of the arithmetic Tutte polynomial.

  • 223.
    Brändén, Petter
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Ottergren, Elin
    A Characterization of Multiplier Sequences for Generalized Laguerre Bases2014Ingår i: Constructive approximation, ISSN 0176-4276, E-ISSN 1432-0940, Vol. 39, nr 3, s. 585-596Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We give a complete characterization of multiplier sequences for generalized Laguerre bases. We also apply our methods to give a short proof of the characterization of Hermite multiplier sequences achieved by Piotrowski.

  • 224.
    Bucher-Karlsson, Michelle
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    On Minimal Triangulations of Products of Convex Polygons2009Ingår i: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 41, nr 2, s. 328-347Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We give new lower bounds for the minimal number of simplices needed in a triangulation of the product of two convex polygons, improving the lower bounds in Bowen et al. (Topology 44:321-339, 2005).

  • 225.
    Bucher-Karlsson, Michelle
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Simplicial volume of locally symmetric spaces covered by SL3R/SO(3)2007Ingår i: Geometriae Dedicata, ISSN 0046-5755, E-ISSN 1572-9168, Vol. 125, nr 1, s. 203-224Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We give the first complete proof of the strict positivity of the simplicial volume of compact locally symmetric spaces covered by SL3R/SO(3) and show why the proof in [Savage RP, Trans Amer Math Soc 274(1):239-263, 1982] is incorrect.

  • 226.
    Bucher-Karlsson, Michelle
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    The simplicial volume of closed manifolds covered by H-2 x H-22008Ingår i: Journal of Topology, ISSN 1753-8416, E-ISSN 1753-8424, Vol. 1, nr 3, s. 584-602Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We compute the value of the simplicial volume for closed, oriented Riemannian manifolds covered by H-2 x H-2 explicitly, and thus in particular for products of closed hyperbolic surfaces. This gives the first exact value of a nonvanishing simplicial volume for a manifold not admitting a hyperbolic structure.

  • 227.
    Bucher-Karlsson, Michelle
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Gelander, Tsachik
    Milnor-Wood inequalities for manifolds locally isometric to a product of hyperbolic planes2008Ingår i: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 346, nr 11-12, s. 661-666Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This Note describes sharp Milnor-Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not admit an affine structure, confirming Chern-Sullivan's conjecture in this case. The manifolds under consideration are of particular interest, since in contrary to some other locally symmetric spaces they do admit interesting flat vector bundles in the corresponding dimension. When the manifold is irreducible and of higher rank, it is shown that flat oriented vector bundles are determined completely by the sign of the Euler number.

  • 228.
    Bylund, Mats
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Stability in Hamiltonian Systems: KAM stability versus instability around an invariant torus2017Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)
    Abstract [sv]

    Under sin ICM-54 föreläsning introducerade Kolmogorov ett numera fundamentalt resultat angående bevarandet av en måtteoretiskt stor mängd invarianta torusar, inom en viss kategori av nästan intagrabla Hamiltonska system. 44 år senare, under sitt ICM-98 tal, formulerade Herman en förmodan om att en invariant diofantisk torus tillhörande en analytisk Hamiltonian alltid omges av en mängd invarianta KAM torusar av positivt mått.

    Detta examensarbete bygger vidare på befintliga resultat och ger i fallet tre frihetsgrader ett motexempel till KAM stabilitet kring en invariant torus, i kategorin glatta Hamiltonska system. Arbetet är självtillräckligt i den mening att det även ges en kort introduktion till Hamiltonska system, samt en exposition av Kolmogorovs klassiska resultat.

  • 229.
    Börjesson, Oscar
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
    HaQ, Sebastian Rezwanul
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Do hedge funds yield greater risk-adjusted rate of  returns than mutual funds?A quantitative study comparing hedge funds to mutual funds and hedge fund strategies2014Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
    Abstract [sv]

    Hedgefonder har den senaste tiden ökat i popularitet. Samtidigt finns det delade meningar huruvida hedgefonder genererar absolutavkastning och om de fungerar som bra alternativ till traditionella fonder. Denna uppsats syftar till att undersöka huruvida hedgefonder skapar absolutavkastning samt om det finns investeringsstrategier som presterar bättre än andra. Denna uppsats jämför hedgefonders riskjusterade avkastning med traditionella fonder, för att på sätt se om en viss investeringsstrategi ar mer lukrativ i termer av överavkastning i förhållande till motsvarande index. Vi har använt ekonometriska metoder för att söka efter statistiskt signifikanta skillnader mellan avkastningen för hedgefonder och traditionella fonder.

    Våra resultat visar att svenska hedgefonder inte genererar högre risk-justerade avkastningar än svenska aktiefonder. Våra resultat visar inga signifikanta skillnader vad gäller avkastning mellan olika strategier. Slutligen finner vi heller inga bevis för att hedgefonder går emot den effektiva marknadshypotesen

  • 230. Canto-Martín, Francisco
    et al.
    Hedenmalm, Håkan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Montes-Rodríguez, Alfonso
    Perron-Frobenius operators and the Klein-Gordon equation2014Ingår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 16, nr 1, s. 31-66Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    For a smooth curve Gamma and a set Lambda in the plane R-2, let AC(Gamma; Lambda) be the space of finite Borel measures in the plane supported on Gamma, absolutely continuous with respect to arc length and whose Fourier transform vanishes on Lambda. Following [12], we say that (Gamma, Lambda) is a Heisenberg uniqueness pair if AC(Gamma; Lambda) = {0}. In the context of a hyperbola Gamma, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets Gamma of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the issue of finding the dimension of AC(Gamma; Lambda) when it is nonzero. We will fix the curve Gamma to be the hyperbola x(1)x(2) = 1, and the set Lambda = Lambda(alpha,beta) to be the lattice-cross Lambda(alpha,beta) = (alpha Zeta x {0}) boolean OR ({0} x beta Z), where alpha, beta are positive reals. We will also consider Gamma(+), the branch of x(1)x(2) = 1 where x(1) > 0. In [12], it is shown that AC(Gamma; Lambda(alpha,beta)) = {0} if and only if alpha beta <= 1. Here, we show that for alpha beta > 1, we get a rather drastic "phase transition": AC(Gamma; Lambda(alpha,beta)) is infinite-dimensional whenever alpha beta > 1. It is shown in [13] that AC(Gamma(+); Lambda(alpha,beta)) = {0} if and only if alpha beta < 4. Moreover, at the edge alpha beta = 4, the behavior is more exotic: the space AC(Gamma(+); Lambda(alpha,beta)) is one-dimensional. Here, we show that the dimension of AC(Gamma(+); Lambda(alpha,beta)) is infinite whenever alpha beta > 4. Dynamical systems, and more specifically Perron-Frobenius operators, play a prominent role in the presentation.

  • 231.
    Carlqvist, Håkan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Multiscale analysis of multi-channel signals2005Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    I: Amplitude and phase relationship between alpha and beta oscillations in the human EEG We have studied the relation between two oscillatory patterns within EEG signals (oscillations with main frequency 10 Hz and 20 Hz), with wavelet-based methods. For better comparison, a variant of the continuous wavelet transform, was derived. As a conclusion, the two patterns were closely related and 70-90 % of the activity in the 20 Hz pattern could be seen as a resonance phenomenon of the 10 Hz activity.

    II: A local discriminant basis algorithm using wavelet packets for discrimination between classes of multidimensional signals We have improved and extended the local discriminant basis algorithm for application on multidimensional signals appearing from multichannels. The improvements includes principal-component analysis and crossvalidation- leave-one out. The method is furthermore applied on two classes of EEG signals, one group of control subjects and one group of subjects with type I diabetes. There was a clear discrimination between the two groups. The discrimination follows known differences in the EEG between the two groups of subjects.

    III: Improved classification of multidimensional signals using orthogonality properties of a time-frequency library We further improve and refine the method in paper2 and apply it on 4 classes of EEG signals from subjects differing in age and/or sex, which are known factors of EEG alterations. As a method for deciding the best basis we derive an orthogonalbasis- pursuit-like algorithm which works statistically better (Tukey's test for simultaneous confidence intervals) than the basis selection method in the original local discriminant basis algorithm. Other methods included were Fisher's class separability, partial-least-squares and cross-validation-leave-one-subject out. The two groups of younger subjects were almost fully discriminated between each other and to the other groups, while the older subjects were harder to discriminate.

  • 232.
    Carlqvist, Håkan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Nikulin, Vadim V.
    Department of Clinical Neuroscience, Karolinska Institutet, Karolinska University Hospital, Stockholm.
    Strömberg, Jan-Olov
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Brismar, T.
    Department of Clinical Neuroscience, Karolinska Institutet, Karolinska University Hospital, Stockholm.
    Amplitude and phase relationship between alpha and beta oscillations in the human electroencephalogram2005Ingår i: Medical and Biological Engineering and Computing, ISSN 0140-0118, E-ISSN 1741-0444, Vol. 43, nr 5, s. 599-607Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The relationship between the electro-encephalographic (EEG) alpha and beta oscillations in the resting condition was investigated in the study. EEGs were recorded in 33 subjects, and alpha (7.5-12.5Hz) and beta (15-25Hz) oscillations were extracted with the use of a modified wavelet transform. Power, peak frequency and phase synchronisation were evaluated for both types of oscillation. The average beta-alpha peak frequency ratio was about 1.9-2.0 for all electrode derivations. The peak frequency of beta activity was within 70-90 % of the 95 % confidence interval of twice the alpha frequency. A significant (p < 0.05) linear regression was found between beta and alpha power in all derivations in 32 subjects, with the slope of the regression line being approximate to 0.3. There was no significant difference in the slope of the line in different electrode locations, although the power correlation was strongest in the occipital locations where alpha and beta oscillations had the largest power. A significant 1:2 phase synchronisation was present between the alpha and beta oscillations, with a phase lag of about pi/2 in all electrode derivations. The strong frequency relationship between the resting beta and alpha oscillations suggests that they are generated by a common mechanism. Power and phase relationships were weaker suggesting that these properties can be modulated by additional mechanisms as well as be influenced by noise. A careful distinction between alpha-dependent and alpha-independent beta activity should be considered when making statements about the possible significance of genuine beta activity in different neurophysiological mechanisms.

  • 233.
    Carlqvist, Håkan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Strömberg, Jan-Olov
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Improved classification of multidimensional signals using orthogonality properties of a time-frequency libraryArtikel i tidskrift (Övrigt vetenskapligt)
  • 234.
    Carlqvist, Håkan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Sundberg, Rolf
    Strömberg, Jan-Olov
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    A local discriminant basis algorithm using wavelet packets for discrimination between classes of mulridimensionals signalsArtikel i tidskrift (Övrigt vetenskapligt)
  • 235.
    Carlson, Magnus
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Abelian affine group schemes, plethories, and arithmetic topology2018Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    In Paper A we classify plethories over a field of characteristic zero. All plethories over characteristic zero fields are ``linear", in the sense that they are free plethories on a bialgebra. For the proof of this classification we need some facts from the theory of ring schemes where we extend previously known results. We also give a classification of plethories with trivial Verschiebung over a perfect field k of characteristic p>0.

    In Paper B we study tensor products of abelian affine group schemes over a perfect field k. We first prove that the tensor product G_1 \otimes G_2 of two abelian affine group schemes G_1,G_2 over a perfect field k exists. We then describe the multiplicative and unipotent part of the group scheme G_1 \otimes G_2. The multiplicative part is described in terms of Galois modules over the absolute Galois group of k. In characteristic zero the unipotent part of G_1 \otimes G_2 is the group scheme whose primitive elements are P(G_1) \otimes P(G_2). In positive characteristic, we give a formula for the tensor product in terms of Dieudonné theory. 

    In Paper C we use ideas from homotopy theory to define new obstructions to solutions of embedding problems and compute the étale cohomology ring of the ring of integers of a totally imaginary number field with coefficients in Z/2Z. As an application of the obstruction-theoretical machinery, we give an infinite family of totally imaginary quadratic number fields such that Aut(PSL(2,q^2)), for q an odd prime power, cannot be realized as an unramified Galois group over K, but its maximal solvable quotient can. 

    In Paper D we compute the étale cohomology ring of an arbitrary number field with coefficients in Z/nZ for n an arbitrary positive integer. This generalizes the computation in Paper C. As an application, we give a formula for an invariant defined by Minhyong Kim.

  • 236.
    Carlson, Magnus
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Classification of plethories in characteristic zero2015Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    We classify plethories over fields of characteristic zero, thus answering a question of Borger-Wieland and Bergman-Hausknecht. All plethories over characteristic zero fields are linear, in the sense that they are free plethories on a bialgebra. For the proof we need some facts from the theory of ring schemes where we extend previously known results. We also classify plethories with trivial Verschiebung over a perfect field of non-zero characteristic and indicate future work. 

  • 237.
    Carlson, Magnus
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Classification of plethories in characteristic zeroManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    We classify plethories over fields of characteristic zero, thus answering a question of Borger-Wieland and Bergman. All plethories over characteristic zero fields are linear, in the sense that they are free plethories on a bialgebra. For the proof we need some facts from the theory of ring schemes where we extend previously known results. We also classify plethories with trivial Verschiebung over a perfect field of non-zero characteristic and indicate future work.

  • 238.
    Carlson, Magnus
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Tensor products of affine and formal abelian groupsManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    In this paper we study tensor products of affine abelian group schemes over a perfect field k. We first prove that the tensor product G_1 ⊗ G_2 of two affine abelian group schemes G_1,G_2  over a perfect field k exists. We then describe the multiplicative and unipotent part of the group scheme G_1 ⊗G_2. The multiplicative part is described in terms of Galois modules over the absolute Galois group of k. We describe the unipotent part of G_1 ⊗ G_2 explicitly, using Dieudonn\'e theory in positive characteristic. We relate these constructions to previously studied tensor products of formal group schemes.

  • 239.
    Carlson, Magnus
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Ahlqvist, Eric
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    The cohomology ring of the ring of integers of a number fieldManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    We compute the étale cohomology ring H^*(Spec O_K,Z/nZ) where O_K is the ring of integers of a number field K. As an application, we give a non-vanishing formula for an invariant defined by Minhyong Kim.

  • 240.
    Carlson, Magnus
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Bauer, Tilman
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Tensor products of affine and formal abelian groupsManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    In this paper we study tensor products of affine abelian group schemes over a perfect field k. We first prove that the tensor product G_1 ⊗ G_2 of two affine abelian group schemes G_1,G_2  over a perfect field k exists. We then describe the multiplicative and unipotent part of the group scheme G_1 ⊗G_2. The multiplicative part is described in terms of Galois modules over the absolute Galois group of k. We describe the unipotent part of G_1 ⊗ G_2 explicitly, using Dieudonn\'e theory in positive characteristic. We relate these constructions to previously studied tensor products of formal group schemes.

  • 241.
    Carlson, Magnus
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Schlank, Tomer
    Hebrew University of Jerusalem.
    The unramified inverse Galois problem and cohomology rings of totally imaginary number fieldsManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    We employ methods from homotopy theory to define new obstructions to solutions of embedding problems. By using these novel obstructions we study embedding problems with non-solvable kernel. We apply these obstructions to study the unramified inverse Galois problem. That is, we show that our methods can be used to determine that certain groups cannot be realized as the Galois groups of unramified extensions of certain number fields. To demonstrate the power of our methods, we give an infinite family of totally imaginary quadratic number fields such that Aut(PSL(2,q^2)) for q an odd prime power, cannot be realized as an unramified Galois group over K, but its maximal solvable quotient can. To prove this result, we determine the ring structure of the \'etale cohomology ring H^*(Spec O_K;Z/2Z) where O_K is the ring of integers of an arbitrary totally imaginary number field K.

  • 242. Casagrande, Cinzia
    et al.
    Di Rocco, Sandra
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Projective Q-factorial toric varieties covered by lines2008Ingår i: Communications in Contemporary Mathematics, ISSN 0219-1997, Vol. 10, nr 3, s. 363-389Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We give a structural theorem for Q-factorial toric varieties covered by lines in P-N, and compute their dual defect. This yields a characterization of defective Q-factorial toric varieties in P-N. The combinatorial description of such varieties is used to characterize some finite sets of monomials with discriminant equal to one.

  • 243. Cattani, Eduardo
    et al.
    Cueto, Maria Angelica
    Dickenstein, Alicia
    Di Rocco, Sandra
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Sturmfels, Bernd
    Mixed discriminants2013Ingår i: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 274, nr 3-4, s. 761-778Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The mixed discriminant of Laurent polynomials in variables is the irreducible polynomial in the coefficients which vanishes whenever two of the roots coincide. The Cayley trick expresses the mixed discriminant as an -discriminant. We show that the degree of the mixed discriminant is a piecewise linear function in the Plucker coordinates of a mixed Grassmannian. An explicit degree formula is given for the case of plane curves.

  • 244. Cautis, Sabin
    et al.
    Logvinenko, Timothy
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    A derived approach to geometric McKay correspondence in dimension three2009Ingår i: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, Vol. 636, s. 193-236Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We propose a three dimensional generalization of the geometric McKay correspondence described by Gonzales-Sprinberg and Verdier in dimension two. We work it out in detail in abelian case. More precisely, we show that the Bridgeland-King-Reid derived category equivalence induces a natural geometric correspondence between irreducible representations of G and subschemes of the exceptional set of G-Hilb (C-3). This correspondence appears to be related to Reid's recipe.

  • 245.
    Chacholski, Wojciech
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Covers of groups2014Ingår i: ALPINE EXPEDITION THROUGH ALGEBRAIC TOPOLOGY, KTH, Dept Math, S-10044 Stockholm, Sweden., 2014, s. 109-131Konferensbidrag (Refereegranskat)
    Abstract [en]

    We discuss constructions and categorical properties of various cellular approximation functors in the category of groups.

  • 246.
    Chacholski, Wojciech
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Damian, E.
    Farjoun, E. D.
    Segev, Y.
    The A-core and A-cover of a group2009Ingår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 321, nr 2, s. 631-666Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This paper provides a comprehensive investigation of the cellular approximation functor cell(A) G, in the category of groups. approximating a group G by a group A. We also study related notions such as A-injection, A-generation and A-constructibility of a group G and we find several interesting connections with the Schur multiplier H-2(G, Z). Our constructions are direct and are given in a slow and detailed manner.

  • 247.
    Chacholski, Wojciech
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Dwyer, W. G.
    Intermont, M.
    nu(*)-torsion spaces and thick classes2006Ingår i: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 336, nr 1, s. 13-26Artikel i tidskrift (Refereegranskat)
  • 248.
    Chacholski, Wojciech
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Lundman, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Ramanujam, Ryan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Scolamiero, Martina
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Öberg, Sebastian
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Multidimensional Persistence and NoiseManuskript (preprint) (Övrigt vetenskapligt)
  • 249.
    Chacholski, Wojciech
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Neeman, Amnon
    Australian Natl Univ, Math Sci Inst, Ctr Math & Its Applicat, Canberra, ACT 0200, Australia..
    Pitsch, Wolfgang
    Univ Autonoma Barcelona, Dept Matemat, Bellaterra 08193, Cerdanyola Del, Spain..
    Scherer, Jerome
    Ecole Polytech Fed Lausanne, Inst Math, Stn 8, CH-1015 Lausanne, Switzerland..
    Relative Homological Algebra Via Truncations2018Ingår i: Documenta Mathematica, ISSN 1431-0635, E-ISSN 1431-0643, Vol. 23, s. 895-937Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spal-tenstein solved this problem for chain complexes of R-modules by truncating further and further to the left, resolving the pieces, and gluing back the partial resolutions. Our aim is to give a homotopy theoretical interpretation of this procedure, which may be extended to a relative setting. We work in an arbitrary abelian category A and fix a class of "injective objects" I. We show that Spaltenstein's construction can be captured by a pair of adjoint functors between unbounded chain complexes and towers of non-positively graded ones. This pair of adjoint functors forms what we call a Quillen pair and the above process of truncations, partial resolutions, and gluing, gives a meaningful way to resolve complexes in a relative setting up to a split error term. In order to do homotopy theory, and in particular to construct a well behaved relative derived category D(A;I), we need more: the split error term must vanish. This is the case when I is the class of all injective R-modules but not in general, not even for certain classes of injectives modules over a Noetherian ring. The key property is a relative analogue of Roos's AB4*-n axiom for abelian categories. Various concrete examples such as Gorenstein homological algebra and purity are also discussed.

  • 250.
    Chacholski, Wojciech
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Pitsch, Wolfgang
    KTH.
    Scherer, Jerome
    KTH.
    Homotopy pull-back squares up to localization2006Ingår i: Alpine Anthology of Homotopy Theory / [ed] Arlettaz, D; Hess, K, PROVIDENCE: AMER MATHEMATICAL SOC , 2006, Vol. 399, s. 55-72Konferensbidrag (Refereegranskat)
    Abstract [en]

    We characterize the class of homotopy pull-back squares by means of elementary closure properties. The so called Puppe theorem which identifies the homotopy fiber of certain maps constructed as homotopy colimits is a straightforward consequence. Likewise we characterize the class of squares which are homotopy pull-backs "up to Bousfield localization". This yields a generalization of Puppe's theorem which allows us to identify the homotopy type of the localized homotopy fiber. When the localization functor is homological localization this is one of the key ingredients in the group completion theorem.

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