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1. Fast multiplication of matrices over a finitely generated semiring Andren, Daniel PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_0_j_idt591",{id:"formSmash:items:resultList:0:j_idt591",widgetVar:"widget_formSmash_items_resultList_0_j_idt591",onLabel:"Andren, Daniel ",offLabel:"Andren, Daniel ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); et al. PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_0_j_idt594",{id:"formSmash:items:resultList:0:j_idt594",widgetVar:"widget_formSmash_items_resultList_0_j_idt594",onLabel:"et al.",offLabel:"et al.",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Umeå University.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:0:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Hellström, LarsUmeå University.Markström, KlasUmeå University.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:0:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Fast multiplication of matrices over a finitely generated semiring2008Inngår i: Information Processing Letters, ISSN 0020-0190, E-ISSN 1872-6119, Vol. 107, nr 6, s. 230-234Artikkel i tidsskrift (Fagfellevurdert)2. A Rewriting Approach to Graph Invariants Hellström, Lars PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_1_j_idt591",{id:"formSmash:items:resultList:1:j_idt591",widgetVar:"widget_formSmash_items_resultList_1_j_idt591",onLabel:"Hellström, Lars ",offLabel:"Hellström, Lars ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Mälardalens högskola, Akademin för utbildning, kultur och kommunikation. Umeå University.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:1:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:1:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); A Rewriting Approach to Graph Invariants2009Inngår i: Generalized Lie Theory in Mathematics, Physics and Beyond / [ed] Sergei Silvestrov, Eugen Paal, Viktor Abramov, Alexander Stolin, Berlin, Heidelberg: Springer, 2009, s. 47-67Kapittel i bok, del av antologi (Fagfellevurdert)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_1_j_idt629_0_j_idt630",{id:"formSmash:items:resultList:1:j_idt629:0:j_idt630",widgetVar:"widget_formSmash_items_resultList_1_j_idt629_0_j_idt630",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); 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”.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:1:j_idt629:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 3. Critical Pairs in Network Rewriting Hellström, Lars PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_2_j_idt591",{id:"formSmash:items:resultList:2:j_idt591",widgetVar:"widget_formSmash_items_resultList_2_j_idt591",onLabel:"Hellström, Lars ",offLabel:"Hellström, Lars ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Mälardalens högskola, Akademin för utbildning, kultur och kommunikation. Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:2:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:2:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Critical Pairs in Network Rewriting2014Inngår i: IWC 2014: 3rd International Workshop on Confluence / [ed] Takahito Aoto & Delia Kesner, 2014, s. 9-13Konferansepaper (Fagfellevurdert)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_2_j_idt629_0_j_idt630",{id:"formSmash:items:resultList:2:j_idt629:0:j_idt630",widgetVar:"widget_formSmash_items_resultList_2_j_idt629_0_j_idt630",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); 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.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:2:j_idt629:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Fulltekst (pdf)fulltext$(function(){PrimeFaces.cw("Tooltip","widget_formSmash_items_resultList_2_j_idt854_0_j_idt857",{id:"formSmash:items:resultList:2:j_idt854:0:j_idt857",widgetVar:"widget_formSmash_items_resultList_2_j_idt854_0_j_idt857",showEffect:"fade",hideEffect:"fade",target:"formSmash:items:resultList:2:j_idt854:0:fullText"});}); 4. Literate sources for content dictionaries Hellström, Lars PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_3_j_idt591",{id:"formSmash:items:resultList:3:j_idt591",widgetVar:"widget_formSmash_items_resultList_3_j_idt591",onLabel:"Hellström, Lars ",offLabel:"Hellström, Lars ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Umeå universitet.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:3:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:3:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Literate sources for content dictionaries2013Inngår 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 , 2013Konferansepaper (Fagfellevurdert)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_3_j_idt629_0_j_idt630",{id:"formSmash:items:resultList:3:j_idt629:0:j_idt630",widgetVar:"widget_formSmash_items_resultList_3_j_idt629_0_j_idt630",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); 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.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:3:j_idt629:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Fulltekst (pdf)fulltext$(function(){PrimeFaces.cw("Tooltip","widget_formSmash_items_resultList_3_j_idt854_0_j_idt857",{id:"formSmash:items:resultList:3:j_idt854:0:j_idt857",widgetVar:"widget_formSmash_items_resultList_3_j_idt854_0_j_idt857",showEffect:"fade",hideEffect:"fade",target:"formSmash:items:resultList:3:j_idt854:0:fullText"});}); 5. Literate Sources for Content Dictionaries Hellström, Lars PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_4_j_idt591",{id:"formSmash:items:resultList:4:j_idt591",widgetVar:"widget_formSmash_items_resultList_4_j_idt591",onLabel:"Hellström, Lars ",offLabel:"Hellström, Lars ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:4:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:4:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Literate Sources for Content Dictionaries: a Progress Report2014Inngår i: CICM-WS-WiP 2014: Workshop and Work in Progress Papers at CICM 2014 / [ed] Matthew England et al., Aachen: CEUR-WS.org , 2014Konferansepaper (Fagfellevurdert)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_4_j_idt629_0_j_idt630",{id:"formSmash:items:resultList:4:j_idt629:0:j_idt630",widgetVar:"widget_formSmash_items_resultList_4_j_idt629_0_j_idt630",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); 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.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:4:j_idt629:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 6. Mathematical Formulae Beyond the Tree Paradigm Hellström, Lars PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_5_j_idt591",{id:"formSmash:items:resultList:5:j_idt591",widgetVar:"widget_formSmash_items_resultList_5_j_idt591",onLabel:"Hellström, Lars ",offLabel:"Hellström, Lars ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:5:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:5:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Mathematical Formulae Beyond the Tree Paradigm2016Konferansepaper (Annet vitenskapelig)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_5_j_idt629_0_j_idt630",{id:"formSmash:items:resultList:5:j_idt629:0:j_idt630",widgetVar:"widget_formSmash_items_resultList_5_j_idt629_0_j_idt630",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); 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.PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:5:j_idt629:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 7. Network Rewriting I Hellström, Lars PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_6_j_idt591",{id:"formSmash:items:resultList:6:j_idt591",widgetVar:"widget_formSmash_items_resultList_6_j_idt591",onLabel:"Hellström, Lars ",offLabel:"Hellström, Lars ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Mälardalens högskola, Akademin för utbildning, kultur och kommunikation.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:6:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:6:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Network Rewriting I: The Foundation2012Rapport (Annet vitenskapelig)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_6_j_idt629_0_j_idt630",{id:"formSmash:items:resultList:6:j_idt629:0:j_idt630",widgetVar:"widget_formSmash_items_resultList_6_j_idt629_0_j_idt630",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); 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.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:6:j_idt629:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Fulltekst (pdf)fulltext$(function(){PrimeFaces.cw("Tooltip","widget_formSmash_items_resultList_6_j_idt854_0_j_idt857",{id:"formSmash:items:resultList:6:j_idt854:0:j_idt857",widgetVar:"widget_formSmash_items_resultList_6_j_idt854_0_j_idt857",showEffect:"fade",hideEffect:"fade",target:"formSmash:items:resultList:6:j_idt854:0:fullText"});}); 8. Network Rewriting II Hellström, Lars PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_7_j_idt591",{id:"formSmash:items:resultList:7:j_idt591",widgetVar:"widget_formSmash_items_resultList_7_j_idt591",onLabel:"Hellström, Lars ",offLabel:"Hellström, Lars ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:7:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:7:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Network Rewriting II: Bi- and Hopf Algebras2015Inngår i: Leibniz International Proceedings in Informatics, LIPIcs, Volume 36 / [ed] Maribel Fernández, Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik , 2015, Vol. 36, s. 194-208Konferansepaper (Fagfellevurdert)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_7_j_idt629_0_j_idt630",{id:"formSmash:items:resultList:7:j_idt629:0:j_idt630",widgetVar:"widget_formSmash_items_resultList_7_j_idt629_0_j_idt630",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); 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.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:7:j_idt629:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 9. Network Rewriting II: Bi- and Hopf Algebras Hellström, Lars PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_8_j_idt591",{id:"formSmash:items:resultList:8:j_idt591",widgetVar:"widget_formSmash_items_resultList_8_j_idt591",onLabel:"Hellström, Lars ",offLabel:"Hellström, Lars ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Mälardalens högskola, Akademin för utbildning, kultur och kommunikation.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:8:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:8:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Network Rewriting II: Bi- and Hopf AlgebrasManuskript (preprint) (Annet vitenskapelig)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_8_j_idt629_0_j_idt630",{id:"formSmash:items:resultList:8:j_idt629:0:j_idt630",widgetVar:"widget_formSmash_items_resultList_8_j_idt629_0_j_idt630",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); 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.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:8:j_idt629:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Fulltekst (pdf)fulltext$(function(){PrimeFaces.cw("Tooltip","widget_formSmash_items_resultList_8_j_idt854_0_j_idt857",{id:"formSmash:items:resultList:8:j_idt854:0:j_idt857",widgetVar:"widget_formSmash_items_resultList_8_j_idt854_0_j_idt857",showEffect:"fade",hideEffect:"fade",target:"formSmash:items:resultList:8:j_idt854:0:fullText"});}); 10. Ordered and Combinatorial Structures for Higher-Dimensional Rewriting Hellström, Lars PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_9_j_idt591",{id:"formSmash:items:resultList:9:j_idt591",widgetVar:"widget_formSmash_items_resultList_9_j_idt591",onLabel:"Hellström, Lars ",offLabel:"Hellström, Lars ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:9:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:9:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Ordered and Combinatorial Structures for Higher-Dimensional Rewriting2016Inngår i: / [ed] Samuel Mimram, 2016Konferansepaper (Fagfellevurdert)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_9_j_idt629_0_j_idt630",{id:"formSmash:items:resultList:9:j_idt629:0:j_idt630",widgetVar:"widget_formSmash_items_resultList_9_j_idt629_0_j_idt630",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); 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.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:9:j_idt629:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 11. Ordering Networks Hellström, Lars PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_10_j_idt591",{id:"formSmash:items:resultList:10:j_idt591",widgetVar:"widget_formSmash_items_resultList_10_j_idt591",onLabel:"Hellström, Lars ",offLabel:"Hellström, Lars ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Mälardalens högskola, Akademin för utbildning, kultur och kommunikation. Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:10:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:10:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Ordering Networks2014Konferansepaper (Fagfellevurdert)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_10_j_idt629_0_j_idt630",{id:"formSmash:items:resultList:10:j_idt629:0:j_idt630",widgetVar:"widget_formSmash_items_resultList_10_j_idt629_0_j_idt630",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); 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.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:10:j_idt629:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Fulltekst (pdf)fulltext$(function(){PrimeFaces.cw("Tooltip","widget_formSmash_items_resultList_10_j_idt854_0_j_idt857",{id:"formSmash:items:resultList:10:j_idt854:0:j_idt857",widgetVar:"widget_formSmash_items_resultList_10_j_idt854_0_j_idt857",showEffect:"fade",hideEffect:"fade",target:"formSmash:items:resultList:10:j_idt854:0:fullText"});}); 12. Quantifiers and n-ary binders Hellström, Lars PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_11_j_idt591",{id:"formSmash:items:resultList:11:j_idt591",widgetVar:"widget_formSmash_items_resultList_11_j_idt591",onLabel:"Hellström, Lars ",offLabel:"Hellström, Lars ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Umeå universitet.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:11:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:11:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Quantifiers and n-ary binders: an OpenMath standard enhancement proposal2013Inngår 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 , 2013Konferansepaper (Fagfellevurdert)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_11_j_idt629_0_j_idt630",{id:"formSmash:items:resultList:11:j_idt629:0:j_idt630",widgetVar:"widget_formSmash_items_resultList_11_j_idt629_0_j_idt630",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); 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.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:11:j_idt629:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Fulltekst (pdf)fulltext$(function(){PrimeFaces.cw("Tooltip","widget_formSmash_items_resultList_11_j_idt854_0_j_idt857",{id:"formSmash:items:resultList:11:j_idt854:0:j_idt857",widgetVar:"widget_formSmash_items_resultList_11_j_idt854_0_j_idt857",showEffect:"fade",hideEffect:"fade",target:"formSmash:items:resultList:11:j_idt854:0:fullText"});}); 13. The eval symbol for axiomatising variadic functions Hellström, Lars PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_12_j_idt591",{id:"formSmash:items:resultList:12:j_idt591",widgetVar:"widget_formSmash_items_resultList_12_j_idt591",onLabel:"Hellström, Lars ",offLabel:"Hellström, Lars ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:12:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:12:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); The eval symbol for axiomatising variadic functions2014Inngår i: CICM-WS-WiP 2014: Workshop and Work in Progress Papers at CICM 2014 / [ed] Matthew England et al., Aachen: CEUR-WS.org , 2014Konferansepaper (Fagfellevurdert)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_12_j_idt629_0_j_idt630",{id:"formSmash:items:resultList:12:j_idt629:0:j_idt630",widgetVar:"widget_formSmash_items_resultList_12_j_idt629_0_j_idt630",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); 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.PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:12:j_idt629:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 14. Valued Custom Skew Fields with Generalised PBW Property from Power Series Construction Hellström, Lars PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_13_j_idt591",{id:"formSmash:items:resultList:13:j_idt591",widgetVar:"widget_formSmash_items_resultList_13_j_idt591",onLabel:"Hellström, Lars ",offLabel:"Hellström, Lars ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:13:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:13:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Valued Custom Skew Fields with Generalised PBW Property from Power Series Construction2016Inngår i: Engineering Mathematics II: Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization / [ed] Sergei Silvestrov; Milica Rancic, Springer, 2016, s. 33-55Kapittel i bok, del av antologi (Fagfellevurdert)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_13_j_idt629_0_j_idt630",{id:"formSmash:items:resultList:13:j_idt629:0:j_idt630",widgetVar:"widget_formSmash_items_resultList_13_j_idt629_0_j_idt630",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); 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.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:13:j_idt629:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 15. Branch Thinning and the Large-Scale, Self-Similar Structure of Trees Hellström, Lars PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_14_j_idt591",{id:"formSmash:items:resultList:14:j_idt591",widgetVar:"widget_formSmash_items_resultList_14_j_idt591",onLabel:"Hellström, Lars ",offLabel:"Hellström, Lars ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); et al. PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_14_j_idt594",{id:"formSmash:items:resultList:14:j_idt594",widgetVar:"widget_formSmash_items_resultList_14_j_idt594",onLabel:"et al.",offLabel:"et al.",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik. Umea Univ, Dept Math & Math Stat, SE-90187 Umeå, Sweden.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:14:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Carlsson, LinusMälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik. Umea Univ, Dept Math & Math Stat, SE-90187 Umea, Sweden.Falster, Daniel S.Univ New South Wales, Evolut & Ecol Res Ctr, Sydney, NSW 2052, Australia.;Univ New South Wales, Sch Biol Earth & Environm Sci, Sydney, NSW 2052, Australia.;Macquarie Univ, Dept Biol Sci, Sydney, NSW 2109, Australia..Westoby, MarkMacquarie Univ, Dept Biol Sci, Sydney, NSW 2109, Australia..Brannstrom, AkeUmea Univ, Dept Math & Math Stat, SE-90187 Umea, Sweden.;Int Inst Appl Syst Anal, Evolut & Ecol Program, A-2361 Laxenburg, Austria..PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:14:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Branch Thinning and the Large-Scale, Self-Similar Structure of Trees2018Inngår i: American Naturalist, ISSN 0003-0147, E-ISSN 1537-5323, Vol. 192, nr 1, s. E37-E47Artikkel i tidsskrift (Fagfellevurdert)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_14_j_idt629_0_j_idt630",{id:"formSmash:items:resultList:14:j_idt629:0:j_idt630",widgetVar:"widget_formSmash_items_resultList_14_j_idt629_0_j_idt630",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); 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.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:14:j_idt629:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 16. Universal Algebra Applied to Hom-Associative Algebras, and More Hellström, Lars PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_15_j_idt591",{id:"formSmash:items:resultList:15:j_idt591",widgetVar:"widget_formSmash_items_resultList_15_j_idt591",onLabel:"Hellström, Lars ",offLabel:"Hellström, Lars ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); et al. PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_15_j_idt594",{id:"formSmash:items:resultList:15:j_idt594",widgetVar:"widget_formSmash_items_resultList_15_j_idt594",onLabel:"et al.",offLabel:"et al.",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:15:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Makhlouf, AbdenacerUniversity of Haute Alsace, LMIA.Silvestrov, SergeiMälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:15:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Universal Algebra Applied to Hom-Associative Algebras, and More2014Inngår i: Algebra, Geometry, and Mathematical Physics: AGMP, Mulhouse, France, October 2011, Springer Berlin/Heidelberg, 2014, s. 157-199Konferansepaper (Fagfellevurdert)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_15_j_idt629_0_j_idt630",{id:"formSmash:items:resultList:15:j_idt629:0:j_idt630",widgetVar:"widget_formSmash_items_resultList_15_j_idt629_0_j_idt630",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); 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.

PrimeFaces.cw("Panel","tryPanel",{id:"formSmash:items:resultList:15:j_idt629:0:abstractPanel",widgetVar:"tryPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); 17. Commuting Elements in Q-Deformed Heisenberg Algebras Hellström, Lars PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_16_j_idt591",{id:"formSmash:items:resultList:16:j_idt591",widgetVar:"widget_formSmash_items_resultList_16_j_idt591",onLabel:"Hellström, Lars ",offLabel:"Hellström, Lars ",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); et al. PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_16_j_idt594",{id:"formSmash:items:resultList:16:j_idt594",widgetVar:"widget_formSmash_items_resultList_16_j_idt594",onLabel:"et al.",offLabel:"et al.",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:16:orgPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Silvestrov, Sergei D.Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.PrimeFaces.cw("Panel","testPanel",{id:"formSmash:items:resultList:16:etAlPanel",widgetVar:"testPanel",toggleable:true,toggleSpeed:500,collapsed:false,toggleOrientation:"vertical",closable:true,closeSpeed:500}); Commuting Elements in Q-Deformed Heisenberg Algebras2000Bok (Fagfellevurdert)Abstract [en] PrimeFaces.cw("SelectBooleanButton","widget_formSmash_items_resultList_16_j_idt629_0_j_idt630",{id:"formSmash:items:resultList:16:j_idt629:0:j_idt630",widgetVar:"widget_formSmash_items_resultList_16_j_idt629_0_j_idt630",onLabel:"Abstract [en]",offLabel:"Abstract [en]",onIcon:"ui-icon-triangle-1-s",offIcon:"ui-icon-triangle-1-e"}); 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 Relations
- Bases 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 Operators - The Diamond Lemma
- Degree Functions and Gradations
*q*-Special Combinatorics

Readership: Researchers, graduate students and undergraduates in algebra and physics.

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