| 1. |
- Ekerstad, Niklas, et al.
(författare)
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Clinical Frailty Scale classes are independently associated with 6-month mortality for patients after acute myocardial infarction
- 2022
-
Ingår i: European Heart Journal. - : Oxford University Press. - 2048-8726 .- 2048-8734. ; 11:2, s. 89-98
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Tidskriftsartikel (refereegranskat)abstract
- Aims: Data on the prognostic value of frailty to guide clinical decision-making for patients with myocardial infarction (MI) are scarce. To analyse the association between frailty classification, treatment patterns, in-hospital outcomes, and 6-month mortality in a large population of patients with MI.Methods and results: An observational, multicentre study with a retrospective analysis of prospectively collected data using the SWEDEHEART registry. In total, 3381 MI patients with a level of frailty assessed using the Clinical Frailty Scale (CFS-9) were included. Of these patients, 2509 (74.2%) were classified as non-vulnerable non-frail (CFS 1–3), 446 (13.2%) were vulnerable non-frail (CFS 4), and 426 (12.6%) were frail (CFS 5–9). Frailty and non-frail vulnerability were associated with worse in-hospital outcomes compared with non-frailty, i.e. higher rates of mortality (13.4% vs. 4.0% vs. 1.8%), cardiogenic shock (4.7% vs. 2.5% vs. 1.9%), and major bleeding (4.5% vs. 2.7% vs. 1.1%) (allP < 0.001), and less frequent use of evidence-based therapies. In Cox regression analyses, frailty was strongly and independently associated with 6-month mortality compared with non-frailty, after adjustment for age, sex, the GRACE risk score components, and other potential risk factors [hazard ratio (HR) 3.32, 95% confidence interval (CI) 2.30–4.79]. A similar pattern was seen for vulnerable non-frail patients (fully adjusted HR 2.07, 95% CI1.41–3.02).Conclusion: Frailty assessed with the CFS was independently and strongly associated with all-cause 6-month mortality, also after comprehensive adjustment for baseline differences in other risk factors. Similarly, non-frail vulnerability was independently associated with higher mortality compared with those with preserved functional ability.
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| 2. |
- Simonsson, Moa, et al.
(författare)
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Temporal trends in bleeding events in acute myocardial infarction : insights from the SWEDEHEART registry
- 2020
-
Ingår i: European Heart Journal. - : Oxford University Press. - 0195-668X .- 1522-9645. ; 41:7, s. 833-843
-
Tidskriftsartikel (refereegranskat)abstract
- AIMS: To describe the time trends of in-hospital and out-of-hospital bleeding parallel to the development of new treatments and ischaemic outcomes over the last 20 years in a nationwide myocardial infarction (MI) population.METHODS AND RESULTS: Patients with acute MI (n = 371 431) enrolled in the SWEDEHEART registry from 1995 until May 2018 were selected and evaluated for in-hospital bleeding and out-of-hospital bleeding events at 1 year. In-hospital bleeding increased from 0.5% to a peak at 2% 2005/2006 and thereafter slightly decreased to a new plateau around 1.3% by the end of the study period. Out-of-hospital bleeding increased in a stepwise fashion from 2.5% to 3.5 % in the middle of the study period and to 4.8% at the end of the study period. The increase in both in-hospital and out-of-hospital bleeding was parallel to increasing use of invasive strategy and adjunctive antithrombotic treatment, dual antiplatelet therapy (DAPT), and potent DAPT, while the decrease in in-hospital bleeding from 2007 to 2010 was parallel to implementation of bleeding avoidance strategies. In-hospital re-infarction decreased from 2.8% to 0.6% and out-of-hospital MI decreased from 12.6% to 7.1%. The composite out-of-hospital MI, cardiovascular death, and stroke decreased in a similar fashion from 18.4% to 9.1%.CONCLUSION: During the last 20 years, the introduction of invasive and more intense antithrombotic treatment has been associated with an increase in bleeding events but concomitant there has been a substantial greater reduction of ischaemic events including improved survival.
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| 3. |
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| 4. |
- Andrén, Daniel, 1973-, et al.
(författare)
-
On the complexity of matrix reduction over finite fields
- 2007
-
Ingår i: Advances in Applied Mathematics. - : Academic Press. - 0196-8858 .- 1090-2074. ; 39:4, s. 428-452
-
Tidskriftsartikel (refereegranskat)abstract
- We study matrix elimination over finite fields, and present an algorithm which is asymptotically faster than the traditional Gauss--Jordan elimination. We also bound the average and worst-case complexity for the problem, proving that our algorithm is close to being optimal, and show related concentration results for random matrices.Next we present the results of a large computational study of the complexities for small matrices and fields. Here we determine the exact distribution of the complexity for matrices from $\mathrm{GL}_{n}(\mathbb{F}_{q})$, with $n$ an $q$ smallFinally we consider an extension of the problems studied for finite fields to finite semifields. We give a conjecture on the behaviour of a natural analogue of $\mathrm{GL}_{n}$ for semifields and prove this for a certain class of semifields.
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| 5. |
- Bäck, Per, 1987-
(författare)
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On Hom-associative Ore Extensions
- 2022
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis, we introduce and study hom-associative Ore extensions. These are non-unital, non-associative, non-commutative polynomial rings in which the associativity condition is “twisted” by an additive group homomorphism. In particular, they are examples of hom-associative algebras, and they generalize the classical non-commutative polynomial rings introduced by Ore known as Ore extensions to the non-unital, hom-associative setting. At the same time, when the twisted associativity condition is null, they also generalize to the general non-unital, non-associative setting. We deduce necessary and sufficient conditions for hom-associative Ore extensions to exist, and construct concrete examples thereof. These include hom-associative generalizations of the quantum plane, the universal enveloping algebra of the two-dimensional non-abelian Lie algebra, and the first Weyl algebra, to name a few. The aforementioned algebras turn out to be formal hom-associative deformations of their associative counterparts, the latter two which cannot be formally deformed in the associative setting. Moreover, these are all weakly unital algebras, and we provide a way of embedding any multiplicative, non-unital hom-associative algebra into a multiplicative, weakly unital hom-associative algebra. This generalizes the classical unitalization of non-unital, associative algebras. We then study the hom-associative Weyl algebras in arbitrary characteristic, classify them up to isomorphism, and in the zero characteristic case, we prove that an analogue of the Dixmier conjecture is true. We also study hom-modules over hom-associative rings, and by doing so, we are able to prove a Hilbert's basis theorem for hom-associative Ore extensions. Our theorem includes as special cases both the classical Hilbert's basis theorem for Ore extensions and a Hilbert's basis theorem for unital, non-associative Ore extensions. Last, we construct examples of both hom-associative and non-associative Ore extensions which are all Noetherian by our theorem.
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| 6. |
- Hellström, Anna-Karin, 1974, et al.
(författare)
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Carbamate Chemistry at Interfaces: Practical Considerations and Challenges of Studying Amine Surfactants
- 2019
-
Ingår i: Journal of Surfactants and Detergents. - : Wiley. - 1558-9293 .- 1097-3958. ; 22:5, s. 1109-1117
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- It is well known that CO2 reacts with primary and secondary amines in aqueous solutions and forms carbamates. This reaction can have consequences when studying the self-assembly of amines. In this article, we discuss the practical challenges when working with an alkyl Y-shaped amine (Y12-amine) and demonstrate how the formation of carbamate species influences the physicochemistry of the amine-based surfactant. A drift in dynamic surface tension was observed for Y12-amine at pH above pKa due to the reaction with the naturally occurring CO2 from the atmosphere. The drift in dynamic surface tension was more pronounced at pH above pKa, than at and below pKa. Furthermore, the drift in dynamic surface tension of Y12-amine at pH 12 also affected the surfactant's critical micelle concentration (CMC). CMC of Y12-amine determined by the pendant drop at the air/water interface was almost five times higher than at the N-2/water interface. The latter result was in agreement with the one determined by monitoring the change in chemical shift of H-1 NMR in a N-2 atmosphere. Moreover, it was also shown that the adsorption of the amine at different interfaces influences carbamate formation.
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| 7. |
- Hellström, Lars, 1974-
(författare)
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A Content Dictionary for In-Object Comments
- 2020
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Ingår i: Mathematical Software – ICMS 2020. - Cham, Switzerland : Springer International Publishing. - 9783030521998 - 9783030522001 ; , s. 473-481
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Konferensbidrag (refereegranskat)abstract
- It is observed that some OpenMath objects may benefit from containing comments. A content dictionary with suitable attribution symbols is proposed. This content dictionary also provides application symbols for constructing comments that are somewhat more than just plain text strings.
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| 8. |
- Hellström, Lars, 1974-
(författare)
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A Rewriting Approach to Graph Invariants
- 2009
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Ingår i: Generalized Lie Theory in Mathematics, Physics and Beyond. - Berlin, Heidelberg : Springer. - 9783540853312 - 9783540853329 ; , s. 47-67
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Bokkapitel (refereegranskat)abstract
- Diagrammatic calculation is a powerful tool that gets near indispensable when one tries to manage some of the newer algebraic structures that have been popping up in the last couple of decades. Concretely, it generalises the underlying structure of expressions to being general graphs, where traditional algebraic notation only supports path- or treelike expressions. This paper demonstrates how to apply the author's Generic Diamond Lemma in diagrammatic calculations, by solving through elementary rewriting techniques the problem of classifying all multigraph invariants satisfying a linear contract—delete recursion. (As expected, this leads one to rediscover the Tutte polynomial, along with some more degenerate invariants.) In addition, a concept of “semigraph” is defined which formalises the concept of a graph-theoretical “gadget”.
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| 9. |
- Hellström, Lars, 1974-, et al.
(författare)
-
Branch Thinning and the Large-Scale, Self-Similar Structure of Trees
- 2018
-
Ingår i: American Naturalist. - : UNIV CHICAGO PRESS. - 0003-0147 .- 1537-5323. ; 192:1, s. E37-E47
-
Tidskriftsartikel (refereegranskat)abstract
- Branch formation in trees has an inherent tendency toward exponential growth, but exponential growth in the number of branches cannot continue indefinitely. It has been suggested that trees balance this tendency toward expansion by also losing branches grown in previous growth cycles. Here, we present a model for branch formation and branch loss during ontogeny that builds on the phenomenological assumption of a branch carrying capacity. The model allows us to derive approximate analytical expressions for the number of tips on a branch, the distribution of growth modules within a branch, and the rate and size distribution of tree wood litter produced. Although limited availability of data makes empirical corroboration challenging, we show that our model can fit field observations of red maple (Acer rubrum) and note that the age distribution of discarded branches predicted by our model is qualitatively similar to an empirically observed distribution of dead and abscised branches of balsam poplar (Populus balsamifera). By showing how a simple phenomenological assumptionthat the number of branches a tree can maintain is limitedleads directly to predictions on branching structure and the rate and size distribution of branch loss, these results potentially enable more explicit modeling of woody tissues in ecosystems worldwide, with implications for the buildup of flammable fuel, nutrient cycling, and understanding of plant growth.
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| 10. |
- Hellström, Lars, 1974-, et al.
(författare)
-
Commuting Elements in q-Deformed Heisenberg Algebras
- 2000
-
Bok (refereegranskat)abstract
- Noncommutative algebras, rings and other noncommutative objects, along with their more classical commutative counterparts, have become a key part of modern mathematics, physics and many other fields. The q-deformed Heisenberg algebras defined by deformed Heisenberg canonical commutation relations of quantum mechanics play a distinguished role as important objects in pure mathematics and in many applications in physics. The structure of commuting elements in an algebra is of fundamental importance for its structure and representation theory as well as for its applications. The main objects studied in this monograph are q-deformed Heisenberg algebras — more specifically, commuting elements in q-deformed Heisenberg algebras.In this book the structure of commuting elements in q-deformed Heisenberg algebras is studied in a systematic way. Many new results are presented with complete proofs. Several appendices with some general theory used in other parts of the book include material on the Diamond lemma for ring theory, a theory of degree functions in arbitrary associative algebras, and some basic facts about q-combinatorial functions over an arbitrary field. The bibliography contains, in addition to references on q-deformed Heisenberg algebras, some selected references on related subjects and on existing and potential applications.The book is self-contained, as far as proofs and the background material are concerned. In addition to research and reference purposes, it can be used in a special course or a series of lectures on the subject or as complementary material to a general course on algebra. Specialists as well as doctoral and advanced undergraduate students in mathematics and physics will find this book useful in their research and study.Contents:Immediate Consequences of the Commutation RelationsBases and Normal Form in H(q) and H(q,J)Degree in and Gradation of H(q,J)Centralisers of Elements in H(q,J)Centralisers of Elements in H(q)Algebraic Dependence of Commuting Elements in H(q) and H(q,J)Representations of H(q,J) by q-Difference OperatorsThe Diamond LemmaDegree Functions and Gradationsq-Special CombinatoricsReadership: Researchers, graduate students and undergraduates in algebra and physics.
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| 11. |
- Hellström, Lars, 1974-
(författare)
-
Critical Pairs in Network Rewriting
- 2014
-
Ingår i: IWC 2014. ; , s. 9-13
-
Konferensbidrag (refereegranskat)abstract
- This extended abstract breifly introduces rewriting of networks (directed acyclic graphs with the extra structure needed to serve as expressions for PROducts and Permutations categories) and describes the critical pairs aspects of this theory. The author's interest in these comes from wanting to do equational reasoning in algebraic theories (such as Hopf algebras) that mix ordinary operations with co-operations; networks then serve as a formalism for expressions.The main message is to point out two phenomena that arise in network rewriting. The first is that of non-convexity of rules, wherein the left hand side of a rule need not be syntactically similar to a symbol in any extension of the underlying signature. The second is one of critical pairs potentially arising where two redexes wrap around each other even when they do not intersect.
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| 12. |
- Hellström, Lars, 1974-
(författare)
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Literate Sources for Content Dictionaries : a Progress Report
- 2014
-
Ingår i: CICM-WS-WiP 2014. - Aachen : CEUR-WS.org.
-
Konferensbidrag (refereegranskat)abstract
- At OM2013, the author suggested and sketched a system that would use LaTeX documents as Literate Programming sources for content dictionaries. This paper reports on the progress that has since been made with this system. One important milestone that has been reached is that valid .ocd files with CDDefinitions, FMPs, CMPs, and Examples are being generated.
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| 13. |
- Hellström, Lars, 1974-
(författare)
-
Literate sources for content dictionaries
- 2013
-
Ingår i: MathUI, OpenMath, PLMMS and ThEdu Workshops and Work in Progress at the Conference on Intelligent Computer Mathematics. - Aachen : Redaktion Sun SITE, Informatik V, RWTH Aachen.
-
Konferensbidrag (refereegranskat)abstract
- It is suggested that a LaTeX document could be used as the Literate Programming source of an OpenMath content dictionary. Several approaches to literate programming are reviewed and a possible implementation is sketched.
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| 14. |
- Hellström, Lars, 1974-
(författare)
-
Mathematical Formulae Beyond the Tree Paradigm
- 2016
-
Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- A restriction in the traditional language of mathematics is that every formula has an underlying tree structure, wherein an expression decomposes into subexpressions that each contribute one result to the whole. Some modern mathematics (especially in higher algebra and differential geometry) is however not so well catered for by this tree paradigm, and would be much better off if this restriction was lifted; it is not that the mathematics requires a generalised formula language, but it becomes a whole lot easier to state and do when you use one.This poster shows the formula language of networks that arises when one relaxes the tree restriction to instead allow an underlying directed acyclic graph (DAG) structure. Evaluation of networks is non-obvious, but can be carried out in any PROP/symocat; conversely, evaluation of networks characterises PROPs in much the same way as evaluation of words characterises monoids. Networks are isomorphic to the abstract index interpretation of Einstein convention tensor expressions, and demonstrate that these can in fact be given a coordinate-free interpretation. Special cases of the network notation that have been invented independently include the Penrose graphical notation and quantum gate arrays.
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| 15. |
- Hellström, Lars, 1974-
(författare)
-
Network Rewriting I : The Foundation
- 2012
-
Rapport (övrigt vetenskapligt/konstnärligt)abstract
- A theory is developed which uses "networks" (directed acyclic graphs with some extra structure) as a formalism for expressions in multilinear algebra. It is shown that this formalism is valid for arbitrary PROPs (short for 'PROducts and Permutations category'), and conversely that the PROP axioms are implicit in the concept of evaluating a network. Ordinary terms and operads constitute the special case that the graph underlying the network is a rooted tree. Furthermore a rewriting theory for networks is developed. Included in this is a subexpression concept for which is given both algebraic and effective graph-theoretical characterisations, a construction of reduction maps from rewriting systems, and an analysis of the obstructions to confluence that can occur. Several Diamond Lemmas for this rewriting theory are given. In addition there is much supporting material on various related subjects. In particular there is a "toolbox" for the construction of custom orders on the free PROP, so that an order can be tailored to suit a specific rewriting system. Other subjects treated are the abstract index notation in a general PROP context and the use of feedbacks (sometimes called traces) in PROPs.
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| 16. |
- Hellström, Lars, 1974-
(författare)
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Network Rewriting II : Bi- and Hopf Algebras
- 2015
-
Ingår i: Leibniz International Proceedings in Informatics, LIPIcs, Volume 36. - Dagstuhl, Germany : Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik. - 9783939897859 ; , s. 194-208
-
Konferensbidrag (refereegranskat)abstract
- Bialgebras and their specialisation Hopf algebras are algebraic structures that challenge traditional mathematical notation, in that they sport two core operations that defy the basic functional paradigm of taking zero or more operands as input and producing one result as output. On the other hand, these peculiarities do not prevent studying them using rewriting techniques, if one works within an appropriate network formalism rather than the traditional term formalism. This paper restates the traditional axioms as rewriting systems, demonstrating confluence in the case of bialgebras and finding the (infinite) completion in the case of Hopf algebras. A noteworthy minor problem solved along the way is that of constructing a quasi-order with respect to which the rules are compatible.
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| 17. |
- Hellström, Lars, 1974-
(författare)
-
Network Rewriting II: Bi- and Hopf Algebras
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Bialgebras and their specialisation Hopf algebras are algebraic structures that challenge traditional mathematical notation, in that they sport two core operations that defy the basic functional paradigm of taking zero or more operands as input and producing one result as output. On the other hand, these peculiarities do not prevent studying them using rewriting techniques, if one works within an appropriate network formalism. This paper restates the traditional axioms as rewriting systems, demonstrating confluence in the case of bialgebras and finding the (infinite) completion in the case of Hopf algebras. A noteworthy minor problem solved along the way is that of constructing a quasi-order with respect to which the rules are compatible.
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| 18. |
- Hellström, Lars, 1974-
(författare)
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Network Rewriting Utility Description
- 2023
-
Ingår i: Non-commutative and Non-associative Algebra and Analysis Structures. - : Springer. - 9783031320088 ; , s. 429-476
-
Konferensbidrag (refereegranskat)abstract
- This chapter describes the author’s computer program for doing network rewriting calculations, in its capacity as a tool used for scientific exploration — more precisely to systematically discover non-obvious consequences of the axioms for various algebraic structures. In particular this program can cope with algebraic structures, such as bi- and Hopf algebras, that mix classical operations with co-operations.
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| 19. |
- Hellström, Lars, 1974-
(författare)
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Ordered and Combinatorial Structures for Higher-Dimensional Rewriting
- 2016
-
Konferensbidrag (refereegranskat)abstract
- In principle, rewriting is the logically pure activity of transforming abstract expressions according to fully formalised rules, but in practice there is a significant interplay between abstract rules, more-or-less concrete interpretations, and a variety of book-keeping devices that all need to fit together if the rewriting process is to yield results. This paper presents elementary realisations of book-keeping and other formalising devices that are useful in higher-dimensional rewriting, with a focus on the "2-dimensional" case (PROPs and other types of monoidal category). In particular, it explains how one may construct a variety of ordering relations on these object that are sensitive to differences in the underlying graph structure of the objects being rewritten. It also shows how the formal feedback operation can be used to handle nonconvex redexes, which is a phenomenon of higher-dimensional rewriting that lacks a counterpart in word or term rewriting.
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| 20. |
- Hellström, Lars, 1974-
(författare)
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Ordering Networks
- 2014
-
Konferensbidrag (refereegranskat)abstract
- This extended abstract discusses the problem of defining quasi-orders that are suitable for use with network rewriting. The author’s primary interest is in using network rewriting as a tool for equational reasoning in algebraic theories with both operations and co-operations.
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| 21. |
- Hellström, Lars, 1974-
(författare)
-
Quantifiers and n-ary binders : an OpenMath standard enhancement proposal
- 2013
-
Ingår i: Proceedings of the MathUI, OpenMath, PLMMS and ThEdu Workshops and Work in Progress at CICMco-located with Conferences on Intelligent Computer Mathematics (CICM 2013). - Aachen : Redaktion Sun SITE, Informatik V, RWTH Aachen.
-
Konferensbidrag (refereegranskat)abstract
- It is proposed that the restriction in the OpenMath standard that an OMBIND element must have exactly three children should be lifted, to support more general binder symbols. The case of logics with generalised quantifiers is described in some detail, since these turn out to not have a natural encoding within OpenMath 2.0, because of precisely this restriction. That restricting quantifiers to a single body should have such consequences is not trivial, but follows from a theorem in the Logic branch of Philosophy.
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| 22. |
- Hellström, Lars, 1974-
(författare)
-
Strong Hom-associativity
- 2020
-
Ingår i: Algebraic Structures and Applications. - Cham : Springer Nature. - 9783030418496 ; , s. 317-337
-
Bokkapitel (refereegranskat)abstract
- The concept of an algebra being hom-associative is examined, and found to allow some awkward complications. A modified concept of strong hom-associativity is introduced to eliminate those quirks. It is proved that the basic “Yau twist” construction of a hom-associative algebra from an associative algebra does in fact produce strongly hom-associative algebras. It is proved that the axioms for a strongly hom-associative algebra yields a confluent rewrite system, and a basis for the free strongly hom-associative algebra is given a finite presentation through a parsing expression grammar.
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| 23. |
- Hellström, Lars, 1974-
(författare)
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The Diamond Lemma for Power Series Algebras
- 2002
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The main result in this thesis is the generalisation of Bergman's diamond lemma for ring theory to power series rings. This generalisation makes it possible to treat problems in which there arise infinite descending chains. Several results in the literature are shown to be special cases of this diamond lemma and examples are given of interesting problems which could not previously be treated. One of these examples provides a general construction of a normed skew field in which a custom commutation relation holds.There is also a general result on the structure of totally ordered semigroups, demonstrating that all semigroups with an archimedean element has a (up to a scaling factor) unique order-preserving homomorphism to the real numbers. This helps analyse the concept of filtered structure. It is shown that whereas filtered structures can be used to induce pretty much any zero-dimensional linear topology, a real-valued norm suffices for the definition of those topologies that have a reasonable relation to the multiplication operation.The thesis also contains elementary results on degree (as of polynomials) functions, norms on algebras (in particular ultranorms), (Birkhoff) orthogonality in modules, and construction of semigroup partial orders from ditto quasiorders.
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| 24. |
- Hellström, Lars, 1974-
(författare)
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The eval symbol for axiomatising variadic functions
- 2014
-
Ingår i: CICM-WS-WiP 2014. - Aachen : CEUR-WS.org.
-
Konferensbidrag (refereegranskat)abstract
- This paper describes (and constitutes the source for!) the proposed list4 OpenMath content dictionary. The main feature in this content dictionary is the eval symbol, which treats a list of values as the list of children of an application element. This may, among other things, be employed to state properties of variadic functions.
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| 25. |
- Hellström, Lars, 1974-, et al.
(författare)
-
Universal Algebra Applied to Hom-Associative Algebras, and More
- 2014
-
Ingår i: Algebra, Geometry, and Mathematical Physics. - Berlin, Heidelberg : Springer Berlin/Heidelberg. - 9783642553608 ; , s. 157-199
-
Konferensbidrag (refereegranskat)abstract
- The purpose of this paper is to discuss the universal algebra theory of hom-algebras. This kind of algebra involves a linear map which twists the usual identities. We focus on hom-associative algebras and hom-Lie algebras for which we review the main results. We discuss the envelopment problem, operads, and the Diamond Lemma; the usual tools have to be adapted to this new situation. Moreover we study Hilbert series for the hom-associative operad and free algebra, and describe them up to total degree equal 8 and 9 respectively.
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| 26. |
- Hellström, Lars, 1974-
(författare)
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Valued Custom Skew Fields with Generalised PBW Property from Power Series Construction
- 2016
-
Ingår i: Engineering Mathematics II. - Cham : Springer. - 9783319421049 - 9783319421056 ; , s. 33-55
-
Bokkapitel (refereegranskat)abstract
- This chapter describes a construction of associative algebras that, despite starting from a commutation relation that the user may customize quite considerably, still manages to produce algebras with a number of useful properties: they have a Poincaré–Birkhoff–Witt type basis, they are equipped with a norm (actually an ultranorm) that is trivial to compute for basis elements, they are topologically complete, and they satisfy their given commutation relation. In addition, parameters can be chosen so that the algebras will in fact turn out to be skew fields and the norms become valuations. The construction is basically that of a power series algebra with given commutation relation, stated to be effective enough that the other properties can be derived. What is worked out in detail here is the case of algebras with two generators, but only the analysis of the commutation relation is specific for that case.
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