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• 1.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Philosophy of mathematics in “La Science et l’Hypothèse”, from Henri Poincaré.2017Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 2.
Columbia Univ, Dept Math, New York, NY 10027 USA.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
On the immersion classes of nearby Lagrangians2016In: Journal of Topology, ISSN 1753-8416, E-ISSN 1753-8424, Vol. 9, no 1, p. 232-244Article in journal (Refereed)

We show that the transfer map on Floer homotopy types associated to an exact Lagrangian embedding is an equivalence. This provides an obstruction to representing isotopy classes of Lagrangian immersions by Lagrangian embeddings, which, unlike previous obstructions, is sensitive to information that cannot be detected by Floer cochains. We show this by providing a concrete computation in the case of spheres.

• 3.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
Statistical models of breast cancer tumour growth for mammography screening data2012Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 4. Abramovic, A.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied mathematics.
General inequalities via isotonic subadditive functionals2007In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 10, no 1, p. 15-28Article in journal (Refereed)

In this manuscript a number of general inequalities for isotonic subadditive functionals on a set of positive mappings are proved and applied. In particular, it is pointed out that these inequalities both unify and generalize some general forms of the Hö̈lder, Popoviciu, Minkowski, Bellman and Power mean inequalities. Also some refinements of some of these results are proved.

• 5.
Alfred Renyi Inst Math, 13-15 Realtanoda U, H-1053 Budapest, Hungary..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Dehn surgeries and rational homology2017In: Algebraic and Geometric Topology, ISSN 1472-2747, E-ISSN 1472-2739, Vol. 17, no 1, p. 487-527Article in journal (Refereed)

We consider the question of which Dehn surgeries along a given knot bound rational homology balls. We use Ozsvath and Szabo's correction terms in Heegaard Floer homology to obtain general constraints on the surgery coefficients. We then turn our attention to the case of integral surgeries, with particular emphasis on positive torus knots. Finally, combining these results with a lattice-theoretic obstruction based on Donaldson's theorem, we classify which integral surgeries along torus knots of the form Tkq 1; q bound rational homology balls.

• 6.
Alfred Renyi Inst Math, Realtanoda Ut 13-15, H-1053 Budapest, Hungary..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. Michigan State Univ, Dept Math, 619 Red Cedar Rd, E Lansing, MI 48824 USA..
Embedding 3-manifolds in spin 4-manifolds2017In: Journal of Topology, ISSN 1753-8416, E-ISSN 1753-8424, Vol. 10, no 2, p. 301-323Article in journal (Refereed)

An invariant of orientable 3-manifolds is defined by taking the minimum n such that a given 3-manifold embeds in the connected sum of n copies of S-2 x S-2, and we call this n the embedding number of the 3-manifold. We give some general properties of this invariant, and make calculations for families of lens spaces and Brieskorn spheres. We show how to construct rational and integral homology spheres whose embedding numbers grow arbitrarily large, and which can be calculated exactly if we assume the 11/8-Conjecture. In a different direction we show that any simply connected 4-manifold can be split along a rational homology sphere into a positive definite piece and a negative definite piece.

• 7. Aczel, Peter
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematical Logic.
Binary refinement implies discrete exponentiation2006In: Studia Logica, Vol. 84, p. 361-368Article in journal (Refereed)
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Sub-Gaussian tail bounds for the width and height of conditioned Galton–Watson trees2013In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 41, no 2, p. 1072-1087Article in journal (Refereed)

We study the height and width of a Galton-Watson tree with offspring distribution xi satisfying E xi = 1, 0 < Var xi < infinity, conditioned on having exactly n nodes. Under this conditioning, we derive sub-Gaussian tail bounds for both the width (largest number of nodes in any level) and height (greatest level containing a node); the bounds are optimal up to constant factors in the exponent. Under the same conditioning, we also derive essentially optimal upper tail bounds for the number of nodes at level k, for 1 <= k <= n.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
On the Spread of Random Graphs2014In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 23, no 4, p. 477-504Article in journal (Refereed)

The spread of a connected graph G was introduced by Alon, Boppana and Spencer [1], and measures how tightly connected the graph is. It is defined as the maximum over all Lipschitz functions f on V(G) of the variance of f(X) when X is uniformly distributed on V(G). We investigate the spread for certain models of sparse random graph, in particular for random regular graphs G(n,d), for Erdos-Renyi random graphs G(n,p) in the supercritical range p > 1/n, and for a `small world' model. For supercritical G(n,p), we show that if p = c/n with c > 1 fixed, then with high probability the spread of the giant component is bounded, and we prove corresponding statements for other models of random graphs, including a model with random edge lengths. We also give lower bounds on the spread for the barely supercritical case when p = (1 + o(1))/n. Further, we show that for d large, with high probability the spread of G(n, d) becomes arbitrarily close to that of the complete graph K-n.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Cocompactness and minimizers for inequalities of Hardy-Sobolev type involving N-Laplacian2010In: NoDEA. Nonlinear differential equations and applications (Printed ed.), ISSN 1021-9722, E-ISSN 1420-9004, Vol. 17, no 4, p. 467-477Article in journal (Refereed)

The paper studies quasilinear elliptic problems in the Sobolev spaces W-1,W-p(Omega), Omega subset of R-N, with p = N, that is, the case of Pohozhaev-Trudinger-Moser inequality. Similarly to the case p < N where the loss of compactness in W-1,W-p(R-N) occurs due to dilation operators u bar right arrow t((N-p)/p)u(tx), t > 0, and can be accounted for in decompositions of the type of Struwe's "global compactness" and its later refinements, this paper presents a previously unknown group of isometric operators that leads to loss of compactness in W-0(1,N) over a ball in R-N. We give a one-parameter scale of Hardy-Sobolev functionals, a "p = N"-counterpart of the Holder interpolation scale, for p > N, between the Hardy functional integral vertical bar u vertical bar(p)/vertical bar x vertical bar(p) dx and the Sobolev functional integral vertical bar u vertical bar(pN/(N-mp)) dx. Like in the case p < N, these functionals are invariant with respect to the dilation operators above, and the respective concentration-compactness argument yields existence of minimizers for W-1,W-N-norms under Hardy-Sobolev constraints.

• 11.
TIFR CAM, PB 6503, Bangalore 560065, Karnataka, India.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Defect of compactness in spaces of bounded variation2016In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 271, no 1, p. 37-48Article in journal (Refereed)

Defect of compactness for non-compact imbeddings of Banach spaces can be expressed in the form of a profile decomposition. Let X be a Banach space continuously imbedded into a Banach space Y, and let D be a group of linear isometric operators on X. A profile decomposition in X, relative to D and Y, for a bounded sequence (x(k))(k is an element of N) subset of X is a sequence (S-k)(k is an element of N), such that (x(k) - S-k)(k is an element of N) is a convergent sequence in Y, and, furthermore, S-k has the particular form S-k = Sigma(n is an element of N)g(k)((n))W((n)) with g(k)((n)) is an element of D and w((n)) is an element of X. This paper extends the profile decomposition proved by Solimini [10] for Sobolev spaces (H) over dot(1,P)(R-N) with 1 < p < N to the non-reflexive case p = 1. Since existence of "concentration profiles" w((n)) relies on weak-star compactness, and the space (H) over dot(1,1) is not a conjugate of a Banach space, we prove a corresponding result for a larger space of functions of bounded variation. The result extends also to spaces of bounded variation on Lie groups.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Hardy inequalities for weighted Dirac operator2010In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 189, no 2, p. 241-251Article in journal (Refereed)

An inequality of Hardy type is established for quadratic forms involving Dirac operator and a weight r(-b) for functions in R-n. The exact Hardy constant c(b) = c(b) (n) is found and generalized minimizers are given. The constant cb vanishes on a countable set of b, which extends the known case n = 2, b = 0 which corresponds to the trivial Hardy inequality in R-2. Analogous inequalities are proved in the case c(b) = 0 under constraints and, with error terms, for a bounded domain.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
On a version of Trudinger-Moser inequality with Möbius shift invariance2010In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 39, no 1-2, p. 203-212Article in journal (Refereed)

The paper raises a question about the optimal critical nonlinearity for the Sobolev space in two dimensions, connected to loss of compactness, and discusses the pertinent concentration compactness framework. We study properties of the improved version of the Trudinger-Moser inequality on the open unit disk B subset of R-2, recently proved by Mancini and Sandeep [g], (Arxiv 0910.0971). Unlike the original Trudinger-Moser inequality, this inequality is invariant with respect to the Mobius automorphisms of the unit disk, and as such is a closer analogy of the critical nonlinearity integral |u|(2)* in the higher dimension than the original Trudinger-Moser nonlinearity.

• 14. Adimurthi, no first name
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
On compactness in the Trudinger-Moser inequality2014In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. 13, no 2, p. 399-416Article in journal (Refereed)

We show that the Moser functional J(u) = integral Omega(e(4 pi u2) - 1) dx on the set B = {u is an element of H-0(1)(Omega) : parallel to del u parallel to(2) <= 1}, where Omega subset of R-2 is a bounded domain, fails to be weakly continuous only in the following exceptional case. Define g(s)w(r) = s(-1/2)w(r(s)) for s > 0. If u(k) -> u in B while lim inf J(u(k)) > J(u), then, with some s(k) -> 0, u(k) = g(sk) [(2 pi)(-1/2) min {1, log1/vertical bar x vertical bar}], up to translations and up to a remainder vanishing in the Sobolev norm. In other words, the weak continuity fails only on translations of concentrating Moser functions. The proof is based on a profile decomposition similar to that of Solimini [16], but with different concentration operators, pertinent to the two-dimensional case.

• 15.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Framtagning av metod för analys av livslängdsdata2017Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis

Husqvarna AB has as of today an extensive research and development department.This department serves to control the active product as well as the upcoming ones.The way that is done is through two different sets of tests. The first one being a longterm endurance test with aimed to unveil the durability of a product. Second and finalsort of test is a more one dimensional one. The aim is to determine different specificunits of interest like for example Newton (N).

Today the R&D department has a great knowledge within normal distributed data andsomewhat less when it comes to the opposite, so called none normal distributed data.When endurance is of interest the likelihood of that to be of the latter sort is morecommon than not. For now no complete method has been appointed to make iteasier to process a situation of this kind. Studying ever unique case individually, bylooking at the data, has been the way to go. This causes an inconsistency in theanalysis and makes it purely based on which individual that has done it. Lastly it mayalso, unintentionally, ignore the large picture of how a product has progressed.

To solve these problems this thesis work was put together to propose and conduct amethod. To form this method was an ongoing process throughout the whole thesisperiod. Ideas and thoughts were put forward to be reviewed and discussed. After aseries of tweaks to steer it towards the overall goal the method was finalized. Themethod that was put forward was firmly tested. Also a wide laboration in what themethod actually meant was done.

The result was a method to be applied on none normal distributed data. This methodhas three parts. The first being the report where everything is embraced. The secondpart is a short manual for an operator to use. Last part is an example where themethod is put to use.

• 16. Agerholm, Troels
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
On selfadjoint functors satisfying polynomial relations2011In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 330, no 1, p. 448-467Article in journal (Refereed)

We study selfadjoint functors acting on categories of finite dimensional modules over finite dimensional algebras with an emphasis on functors satisfying some polynomial relations. Selfadjoint functors satisfying several easy relations, in particular, idempotents and square roots of a sum of identity functors. are classified. We also describe various natural constructions for new actions using external direct sums, external tensor products. Serre subcategories, quotients and centralizer subalgebras.

• 17.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Uppsala Univ, Dept Math, SE-75106 Uppsala, Sweden..
Univ Oxford, Dept Stat, Oxford OX1 3TG, England.. IMPA, Rio De Janeiro, RJ, Brazil.. Univ Geneva, Dept Math, Geneva, Switzerland..
Quenched Voronoi percolation2016In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 286, p. 889-911Article in journal (Refereed)

We prove that the probability of crossing a large square in quenched Voronoi percolation converges to 1/2 at criticality, confirming a conjecture of Benjamini, Kalai and Schramm from 1999. The main new tools are a quenched version of the box-crossing property for Voronoi percolation at criticality, and an Efron Stein type bound on the variance of the probability of the crossing event in terms of the sum of the squares of the influences. As a corollary of the proof, we moreover obtain that the quenched crossing event at criticality is almost surely noise sensitive.

• 18.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Inst Nacl Matemat Pura & Aplicada, Estr Dona Castorina 110, BR-22460320 Rio De Janeiro, Brazil.;Uppsala Univ, Dept Math, SE-75106 Uppsala, Sweden..
Univ Gothenburg, Chalmers Univ Technol, Math Sci, SE-41296 Gothenburg, Sweden.. Hungarian Acad Sci, Renyi Inst, 13-15 Realtanoda U, H-1053 Budapest, Hungary.;Budapest Univ Technol & Econ, Inst Math, 1 Egry Jozsef U, H-1111 Budapest, Hungary..
Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome?2017In: Annales de l'I.H.P. Probabilites et statistiques, ISSN 0246-0203, E-ISSN 1778-7017, Vol. 53, no 4, p. 2135-2161Article in journal (Refereed)

Consider a monotone Boolean function f : {0, 1}(n) -> {0, 1} and the canonical monotone coupling {eta(p) : p is an element of [0, 1]} of an element in {0, 1}(n) chosen according to product measure with intensity p is an element of [0, 1]. The random point p is an element of [0, 1] where f (eta(p)) flips from 0 to 1 is often concentrated near a particular point, thus exhibiting a threshold phenomenon. For a sequence of such Boolean functions, we peer closely into this threshold window and consider, for large n, the limiting distribution (properly normalized to be nondegenerate) of this random point where the Boolean function switches from being 0 to 1. We determine this distribution for a number of the Boolean functions which are typically studied and pay particular attention to the functions corresponding to iterated majority and percolation crossings. It turns out that these limiting distributions have quite varying behavior. In fact, we show that any nondegenerate probability measure on R arises in this way for some sequence of Boolean functions.

• 19.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Studier av mätdataregistrering för JAS 39 Gripen2002Independent thesis Advanced level (degree of Master (One Year)), 20 credits / 30 HE creditsStudent thesis
• 20.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
A zero-one law for l-colourable structures with a vectorspace pregeometry2012Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 21.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Combinatorial geometries in model theory2009Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

Model theory and combinatorial pregeometries are closely related throughthe so called algebraic closure operator on strongly minimal sets. Thestudy of projective and ane pregeometries are especially interestingsince they have a close relation to vectorspaces. In this thesis we willsee how the relationship occur and how model theory can concludea very strong classi cation theorem which divides pregeometries withcertain properties into projective, ane and degenerate (trivial) cases.

• 22.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Homogenizable structures and model completeness2016In: Archive for mathematical logic, ISSN 0933-5846, E-ISSN 1432-0665, Vol. 55, no 7-8, p. 977-995Article in journal (Refereed)

A homogenizable structure M is a structure where we may add a finite amount of new relational symbols to represent some 0-definable relations in order to make the structure homogeneous. In this article we will divide the homogenizable structures into different classes which categorize many known examples and show what makes each class important. We will show that model completeness is vital for the relation between a structure and the amalgamation bases of its age and give a necessary and sufficient condition for an countably categorical model-complete structure to be homogenizable.

• 23.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
>k-homogeneous infinite graphs2018In: Journal of combinatorial theory. Series B (Print), ISSN 0095-8956, E-ISSN 1096-0902, Vol. 128, p. 160-174Article in journal (Refereed)

In this article we give an explicit classification for the countably infinite graphs G which are, for some k, ≥k-homogeneous. It turns out that a ≥k  -homogeneous graph M is non-homogeneous if and only if it is either not 1-homogeneous or not 2-homogeneous, both cases which may be classified using ramsey theory.

The full text will be freely available from 2019-09-05 16:29
• 24.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Limit Laws, Homogenizable Structures and Their Connections2018Doctoral thesis, comprehensive summary (Other academic)

This thesis is in the field of mathematical logic and especially model theory. The thesis contain six papers where the common theme is the Rado graph R. Some of the interesting abstract properties of R are that it is simple, homogeneous (and thus countably categorical), has SU-rank 1 and trivial dependence. The Rado graph is possible to generate in a probabilistic way. If we let K be the set of all finite graphs then we obtain R as the structure which satisfy all properties which hold with assymptotic probability 1 in K. On the other hand, since the Rado graph is homogeneous, it is also possible to generate it as a Fraïssé-limit of its age.

Paper I studies the binary structures which are simple, countably categorical, with SU-rank 1 and trivial algebraic closure. The main theorem shows that these structures are all possible to generate using a similar probabilistic method which is used to generate the Rado graph. Paper II looks at the simple homogeneous structures in general and give certain technical results on the subsets of SU-rank 1.

Paper III considers the set K consisting of all colourable structures with a definable pregeometry and shows that there is a 0-1 law and almost surely a unique definable colouring. When generating the Rado graph we almost surely have only rigid structures in K. Paper IV studies what happens if the structures in K are only the non-rigid finite structures. We deduce that the limit structures essentially try to stay as rigid as possible, given the restriction, and that we in general get a limit law but not a 0-1 law.

Paper V looks at the Rado graph's close cousin the random t-partite graph and notices that this structure is not homogeneous but almost homogeneous. Rather we may just add a definable binary predicate, which hold for any two elemenets which are in the same part, in order to make it homogeneous. This property is called being homogenizable and in Paper V we do a general study of homogenizable structures. Paper VI conducts a special case study of the homogenizable graphs which are the closest to being homogeneous, providing an explicit classification of these graphs.

1. Simple structures axiomatized by almost sure theories
Open this publication in new window or tab >>Simple structures axiomatized by almost sure theories
2016 (English)In: Annals of Pure and Applied Logic, ISSN 0168-0072, E-ISSN 1873-2461, Vol. 167, no 5, p. 435-456Article in journal (Refereed) Published
##### Abstract [en]

In this article we give a classification of the binary, simple, ω-categorical structures with SU-rank 1 and trivial algebraic closure. This is done both by showing that they satisfy certain extension properties, but also by noting that they may be approximated by the almost sure theory of some sets of finite structures equipped with a probability measure. This study give results about general almost sure theories, but also considers certain attributes which, if they are almost surely true, generate almost sure theories with very specific properties such as ω-stability or strong minimality.

##### Keywords
Random structure, Almost sure theory, Pregeometry, Supersimple, Countably categorical
##### National Category
Algebra and Logic
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-276995 (URN)10.1016/j.apal.2016.02.001 (DOI)000372680500001 ()
Available from: 2016-03-01 Created: 2016-02-17 Last updated: 2017-11-30Bibliographically approved
2. On sets with rank one in simple homogeneous structures
Open this publication in new window or tab >>On sets with rank one in simple homogeneous structures
2015 (English)In: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 228, p. 223-250Article in journal (Refereed) Published
##### Abstract [en]

We study definable sets D of SU-rank 1 in Meq, where M is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such D can be seen as a 'canonically embedded structure', which inherits all relations on D which are definable in Meq, and has no other definable relations. Our results imply that if no relation symbol of the language of M has arity higher than 2, then there is a close relationship between triviality of dependence and D being a reduct of a binary random structure. Somewhat more precisely: (a) if for every n≥2, every n-type p(x1,...,xn) which is realized in D is determined by its sub-2-types q(xi,xj)⊆p, then the algebraic closure restricted to D is trivial; (b) if M has trivial dependence, then D is a reduct of a binary random structure.

##### Keywords
model theory, homogeneous structure, simple theory, pregeometry, rank, reduct, random structure
##### National Category
Algebra and Logic
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-243006 (URN)10.4064/fm228-3-2 (DOI)000352858400002 ()
Available from: 2015-02-03 Created: 2015-02-03 Last updated: 2017-12-05Bibliographically approved
3. Random l-colourable structures with a pregeometry
Open this publication in new window or tab >>Random l-colourable structures with a pregeometry
2017 (English)In: Mathematical logic quarterly, ISSN 0942-5616, E-ISSN 1521-3870, Vol. 63, no 1-2, p. 32-58Article in journal (Refereed) Published
##### Abstract [en]

We study finite -colourable structures with an underlying pregeometry. The probability measure that is usedcorresponds to a process of generating such structures by which colours are first randomly assigned to all1-dimensional subspaces and then relationships are assigned in such a way that the colouring conditions aresatisfied but apart from this in a random way. We can then ask what the probability is that the resulting structure,where we now forget the specific colouring of the generating process, has a given property. With this measurewe get the following results: (1) A zero-one law. (2) The set of sentences with asymptotic probability 1 has anexplicit axiomatisation which is presented. (3) There is a formula ξ (x, y) (not directly speaking about colours)such that, with asymptotic probability 1, the relation “there is an -colouring which assigns the same colourto x and y” is defined by ξ (x, y). (4) With asymptotic probability 1, an -colourable structure has a unique-colouring (up to permutation of the colours).

##### Place, publisher, year, edition, pages
Wiley-VCH Verlagsgesellschaft, 2017
##### National Category
Algebra and Logic
##### Research subject
Mathematical Logic
##### Identifiers
urn:nbn:se:uu:diva-321515 (URN)10.1002/malq.201500006 (DOI)000400361900003 ()
Available from: 2017-05-06 Created: 2017-05-06 Last updated: 2017-11-28Bibliographically approved
4. Limit laws and automorphism groups of random nonrigid structures
Open this publication in new window or tab >>Limit laws and automorphism groups of random nonrigid structures
2015 (English)In: Journal of Logic and Analysis, ISSN 1759-9008, E-ISSN 1759-9008, Vol. 7, no 2, p. 1-53, article id 1Article in journal (Refereed) Published
##### Abstract [en]

A systematic study is made, for an arbitrary finite relational language with at least one symbol of arity at least 2, of classes of nonrigid finite structures. The well known results that almost all finite structures are rigid and that the class of finite structures has a zero-one law are, in the present context, the first layer in a hierarchy of classes of finite structures with increasingly more complex automorphism groups. Such a hierarchy can be defined in more than one way. For example, the kth level of the hierarchy can consist of all structures having at least k elements which are moved by some automorphism. Or we can consider, for any finite group G, all finite structures M such that G is a subgroup of the group of automorphisms of M; in this case the "hierarchy" is a partial order. In both cases, as well as variants of them, each "level" satisfies a logical limit law, but not a zero-one law (unless k = 0 or G is trivial). Moreover, the number of (labelled or unlabelled) n-element structures in one place of the hierarchy divided by the number of n-element structures in another place always converges to a rational number or to infinity as n -> infinity. All instances of the respective result are proved by an essentially uniform argument.

##### Keywords
finite model theory, limit law, zero-one law, random structure, automorphism group
##### National Category
Algebra and Logic
##### Research subject
Mathematical Logic
##### Identifiers
urn:nbn:se:uu:diva-248078 (URN)10.4115/jla.2015.7.2 (DOI)000359802400001 ()
Available from: 2015-03-26 Created: 2015-03-26 Last updated: 2017-12-04Bibliographically approved
5. Homogenizable structures and model completeness
Open this publication in new window or tab >>Homogenizable structures and model completeness
2016 (English)In: Archive for mathematical logic, ISSN 0933-5846, E-ISSN 1432-0665, Vol. 55, no 7-8, p. 977-995Article in journal (Refereed) Published
##### Abstract [en]

A homogenizable structure M is a structure where we may add a finite amount of new relational symbols to represent some 0-definable relations in order to make the structure homogeneous. In this article we will divide the homogenizable structures into different classes which categorize many known examples and show what makes each class important. We will show that model completeness is vital for the relation between a structure and the amalgamation bases of its age and give a necessary and sufficient condition for an countably categorical model-complete structure to be homogenizable.

##### Keywords
Homogenizable, Model-complete, Amalgamation class, Quantifier-elimination
##### National Category
Algebra and Logic
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-303714 (URN)10.1007/s00153-016-0507-6 (DOI)000385155700010 ()
Available from: 2016-09-22 Created: 2016-09-22 Last updated: 2017-11-28Bibliographically approved
6. >k-homogeneous infinite graphs
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2018 (English)In: Journal of combinatorial theory. Series B (Print), ISSN 0095-8956, E-ISSN 1096-0902, Vol. 128, p. 160-174Article in journal (Refereed) Published
##### Abstract [en]

In this article we give an explicit classification for the countably infinite graphs G which are, for some k, ≥k-homogeneous. It turns out that a ≥k  -homogeneous graph M is non-homogeneous if and only if it is either not 1-homogeneous or not 2-homogeneous, both cases which may be classified using ramsey theory.

##### Keywords
>k-homomogeneous, countably infinite graph
##### National Category
Algebra and Logic
##### Research subject
Mathematical Logic
##### Identifiers
urn:nbn:se:uu:diva-329362 (URN)10.1016/j.jctb.2017.08.007 (DOI)000417771100009 ()
Available from: 2017-09-26 Created: 2017-09-26 Last updated: 2018-04-05Bibliographically approved
• 25.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Simple structures axiomatized by almost sure theories2016In: Annals of Pure and Applied Logic, ISSN 0168-0072, E-ISSN 1873-2461, Vol. 167, no 5, p. 435-456Article in journal (Refereed)

In this article we give a classification of the binary, simple, ω-categorical structures with SU-rank 1 and trivial algebraic closure. This is done both by showing that they satisfy certain extension properties, but also by noting that they may be approximated by the almost sure theory of some sets of finite structures equipped with a probability measure. This study give results about general almost sure theories, but also considers certain attributes which, if they are almost surely true, generate almost sure theories with very specific properties such as ω-stability or strong minimality.

• 26.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
To infinity and back: Logical limit laws and almost sure theories2014Licentiate thesis, comprehensive summary (Other academic)
• 27.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Limit laws and automorphism groups of random nonrigid structures2015In: Journal of Logic and Analysis, ISSN 1759-9008, E-ISSN 1759-9008, Vol. 7, no 2, p. 1-53, article id 1Article in journal (Refereed)

A systematic study is made, for an arbitrary finite relational language with at least one symbol of arity at least 2, of classes of nonrigid finite structures. The well known results that almost all finite structures are rigid and that the class of finite structures has a zero-one law are, in the present context, the first layer in a hierarchy of classes of finite structures with increasingly more complex automorphism groups. Such a hierarchy can be defined in more than one way. For example, the kth level of the hierarchy can consist of all structures having at least k elements which are moved by some automorphism. Or we can consider, for any finite group G, all finite structures M such that G is a subgroup of the group of automorphisms of M; in this case the "hierarchy" is a partial order. In both cases, as well as variants of them, each "level" satisfies a logical limit law, but not a zero-one law (unless k = 0 or G is trivial). Moreover, the number of (labelled or unlabelled) n-element structures in one place of the hierarchy divided by the number of n-element structures in another place always converges to a rational number or to infinity as n -> infinity. All instances of the respective result are proved by an essentially uniform argument.

• 28.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
On sets with rank one in simple homogeneous structures2015In: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 228, p. 223-250Article in journal (Refereed)

We study definable sets D of SU-rank 1 in Meq, where M is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such D can be seen as a 'canonically embedded structure', which inherits all relations on D which are definable in Meq, and has no other definable relations. Our results imply that if no relation symbol of the language of M has arity higher than 2, then there is a close relationship between triviality of dependence and D being a reduct of a binary random structure. Somewhat more precisely: (a) if for every n≥2, every n-type p(x1,...,xn) which is realized in D is determined by its sub-2-types q(xi,xj)⊆p, then the algebraic closure restricted to D is trivial; (b) if M has trivial dependence, then D is a reduct of a binary random structure.

• 29.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Random l-colourable structures with a pregeometry2017In: Mathematical logic quarterly, ISSN 0942-5616, E-ISSN 1521-3870, Vol. 63, no 1-2, p. 32-58Article in journal (Refereed)

We study finite -colourable structures with an underlying pregeometry. The probability measure that is usedcorresponds to a process of generating such structures by which colours are first randomly assigned to all1-dimensional subspaces and then relationships are assigned in such a way that the colouring conditions aresatisfied but apart from this in a random way. We can then ask what the probability is that the resulting structure,where we now forget the specific colouring of the generating process, has a given property. With this measurewe get the following results: (1) A zero-one law. (2) The set of sentences with asymptotic probability 1 has anexplicit axiomatisation which is presented. (3) There is a formula ξ (x, y) (not directly speaking about colours)such that, with asymptotic probability 1, the relation “there is an -colouring which assigns the same colourto x and y” is defined by ξ (x, y). (4) With asymptotic probability 1, an -colourable structure has a unique-colouring (up to permutation of the colours).

• 30.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Classifying Categories: The Jordan-Hölder and Krull-Schmidt-Remak Theorems for Abelian Categories2018Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 31.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Limitless Analysis2014Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 32.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
Brownian Motions and Scaling Limits of Random Trees2011Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 33.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Monte Carlo Methods in American Put Option Pricing2011Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 34.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Equation Solving in Indian Mathematics2018Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 35.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Analytical and Numerical Study of the Poincaré Map with Applications on the Computation of Periodic Orbits2015Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 36.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Bond Pricing in Stochastic Volatility Models2008Independent thesis Advanced level (degree of Master (One Year)), 20 credits / 30 HE creditsStudent thesis
• 37.
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
An interesting class of operators with unusual Schatten-von Neumann behavior2002In: Function Spaces, Interpolation Theory and Related Topics (Proceedings of the International Conference in honour of Jaak Peetre on his 65th birthday, Lund 2000), p. 61-150Article in journal (Refereed)
• 38.
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
An interesting class of operators with unusual Schatten-von Neumann behavior2001Report (Other scientific)
• 39.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Approximation of pluricomplex Green functions based on Monte Carlo integrationManuscript (preprint) (Other academic)
• 40.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. Albaha University, Faculty of Science, Department of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Probabilistic approximation of partly filled-in composite Julia sets2017In: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 119, no 3, p. 203-220Article in journal (Refereed)

We study properties of the metric space of pluriregular sets and of contractions on that space induced by finite families of proper polynomial mappings of several complex variables. In particular, we show that closed balls in the space of pluriregular sets do not have to be compact and we give a simple proof of applicability of the so-called chaos game in the case of composite Julia sets. Part of the construction of those sets also leads to a computationally viable approximation by simpler sets based on Monte-Carlo simulation.

• 41.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. Jagiellonian University, Faculty of Mathematics and Computer ScienceInstitute of Mathematics, Institute of Mathematics..
Attractors of compactly generated semigroups of regular polynomial mappings.Manuscript (preprint) (Other academic)
• 42.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Mathematics..
Bergman Functions and Nearly Orthonormal PolynomialsManuscript (preprint) (Other academic)
• 43.
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics. Matematisk statistik.
Approximation and Simulation of the Distributions of Scan Statistics for Poisson Processes in Higher Dimensions1998In: Extremes, Vol. 1, p. 111-126Article in journal (Refereed)

Given a Poisson process in two or three dimensions we are interested in the scan statistic, i.e. the largest number of points contained in a translate of a fixed scanning set restricted to lie inside a rectangular area.

The distribution of the scan statistic is accurately approximated for rectangular scanning sets, using a technique that is also extended to higher dimensions.

The accuracy of the approximation is checked through simulation.

• 44.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
Monotonicity of the difference between median and mean of Gamma distributions and of a related Ramanujan sequence2002Report (Other academic)
• 45.
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics. Matematisk statistik.
Monotonicity of the difference between median and mean of gamma distributions and of a related Ramanujan sequence2003In: Bernoulli, ISSN 1350-7265, Vol. 9, no 2, p. 351-371Article in journal (Refereed)

For $n\ge0$, let $\lambda_n$ be the median of the $\Gamma(n+1,1)$ distribution. We prove that the sequence $\{\alpha_n=\lambda_n-n\}$ decreases from $\log 2$ to $2/3$ as $n$ increases from 0 to $\infty$. The difference, $1-\alpha_n$, between the mean and the median thus increases from $1-\log 2$ to $1/3$.

This result also proves the following conjecture by Chen \& Rubin about the Poisson distributions: Let $Y_{\mu}\sim\text{Poisson}(\mu)$, and \lambda_n$be the largest$\mu$such that$P(Y_{\mu}\le n)=1/2$, then$\lambda_n-n$is decreasing in$n$. The sequence$\{\alpha_n\}$is related to a sequence$\{\theta_n\}$, introduced by Ramanujan, which is known to be decreasing and of the form$\theta_n=\frac13+\frac4{135(n+k_n)}$, where$\frac2{21}<k_n\le\frac8{45}$. We also show that the sequence$\{k_n\}\$ is decreasing.

• 46.
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
On measures of average degree for lattices2003Report (Other scientific)
• 47.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
On Measures of Average Degree for Lattices2006In: Combinatorics, Probability and Computing, Vol. 15, no 4, p. 477-488Article in journal (Refereed)

The usual definition of average degree for a non-regular lattice has the disadvantage that it takes the same value for many lattices with clearly different connectivity. We introduce an alternative definition of average degree, which better separates different lattices.

These measures are compared on a class of lattices and are analyzed using a Markov chain describing a random walk on the lattice. Using the new measure, we conjecture the order of both the critical probabilities for bond percolation and the connective constants for self-avoiding walks on these lattices.

• 48.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
On monoticity of some binomial probabilities2010Report (Other academic)
• 49.
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
Upper and Lower Bounds for the Connective Constants of Self-avoiding Walks on the Archimedean and Laves Lattices2005In: J. Phys. A: Math. Gen., no 38, p. 2055-2080Article in journal (Refereed)
• 50.
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
Upper and Lower Bounds for the Connective Constants of Self-avoiding Walks on the Archimedean and Laves Lattices2004Report (Other scientific)
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