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  • 1.
    Björkholdt, Elise
    et al.
    University West, Department of Technology, Mathematics and Computer Science.
    Lundström, Patrik
    University West, Department of Technology, Mathematics and Computer Science. University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Generalizations of the normal basis theorem2004In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 94, p. 5p. 185-190Article in journal (Refereed)
  • 2.
    Cala, Juan
    et al.
    Escuela de Matemáticas, Universidad Industrial de Santander, Bucaramanga (COL).
    Lundström, Patrik
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Pinedo, Hector
    Escuela de Matemáticas, Universidad Industrial de Santander, Bucaramanga (COL).
    Graded modules over object-unital groupoid graded rings2021In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 50, no 2, p. 444-462Article in journal (Refereed)
    Abstract [en]

    In this article, we analyze the category(Formula presented) of unitary G-graded modules over object unital G -graded rings R, being G a groupoid. Here we consider the forgetful functor  G - R- mod and determine many properties (Formula presented.) for which the following implications are valid for modules M in (Formula presented.) M is (Formula presented.) (Formula presented.) U(M) is (Formula presented.) U(M) is (Formula presented.) (Formula presented.) M is (Formula presented.) We treat the cases when (Formula presented.) is any of the properties: direct summand, projective, injective, free and semisimple. Moreover, graded versions of results concerning classical module theory are established, as well as some structural properties (Formula presented.). 

  • 3.
    Cala, Juan
    et al.
    Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, Colombia (COL).
    Lundström, Patrik
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Pinedo, Hector
    Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, Colombia (COL).
    Object-unital groupoid graded rings, crossed products and separability2021In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 44, no 4, p. 1676-1696Article in journal (Refereed)
    Abstract [en]

    We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal) field extensions. This leads us naturally to consider non-unital groupoid graded rings of a particular type that we call object unital. We determine when such rings are strongly graded, crossed products, skew groupoid rings and twisted groupoid rings. We also obtain necessary and sufficient criteria for when object unital groupoid graded rings are separable over their principal component, thereby generalizing previous results from the unital case to a non-unital situation. © 2020 The Author(s). Published with license by Taylor and Francis Group, LLC.

  • 4.
    Darpö, Erik
    et al.
    Nagoya University, Graduate School of Mathematics, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan (JPN).
    Nystedt, Patrik
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Von-Neumann Finiteness and Reversibility in some Classes of Non-Associative Algebras2021In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 24, p. 1245-1258Article in journal (Refereed)
    Abstract [en]

    We investigate criteria for von-Neumann finiteness and reversibility in some classes of non-associative algebras. Types of algebras that are studied include alternative, flexible, quadratic and involutive algebras, as well as algebras obtained by the Cayley–Dickson doubling process. Our results include precise criteria for von-Neumann finiteness and reversibility of involutive algebras in terms of isomorphism types of their 3-dimensional subalgebras. © 2020, The Author(s).

  • 5.
    Lundström, Patrik
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Algebra, trigonometri och analys2023 (ed. 1)Book (Other academic)
  • 6.
    Lundström, Patrik
    University West, Department of Technology, Mathematics and Computer Science.
    Brauer group rings2004In: JP Journal of Algebra, Number Theory & Applications, Vol. 4, p. 22p. 465-494Article in journal (Refereed)
  • 7.
    Lundström, Patrik
    University West, Department of Technology.
    Cohomology and Self-dual Normal Bases for Infinite Galois Field Extensions2002In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 256, p. 531-541Article in journal (Refereed)
    Abstract [en]

    We generalize an injectivity result obtained by Bayer-Fluckiger and Lenstra concerning pointed cohomology sets, defined by norm-one groups of finite-dimensional algebras with involution over fields k of characteristic different from 2, to the case of inverse limits of finite-dimensional k-algebras with involution. We use this generalization to obtain a result about self-dual normal bases for infinite Galois field extensions.

  • 8.
    Lundström, Patrik
    University West, Department of Technology, Mathematics and Computer Science, Division for Mathematics and Sciences.
    Cohomology and the normal basis theorem2007In: Indagationes mathematicae, ISSN 0019-3577, E-ISSN 1872-6100, Vol. 18, p. 69-72Article in journal (Refereed)
  • 9.
    Lundström, Patrik
    University West, Department of Technology, Mathematics and Computer Science.
    Crossed product algebras defined by separable extensions2005In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 283, p. 14p. 723-737Article in journal (Refereed)
  • 10.
    Lundström, Patrik
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Double Calculus2023In: Surveys in Mathematics and its Applications, ISSN 1843-7265, E-ISSN 1842-6298, Vol. 18, p. 27-48Article in journal (Refereed)
    Abstract [en]

    We present a streamlined, slightly modified version, in the two-variable situation, of a beautiful, but not so well known, theory by Bögel [1, 2], already from the 1930s, on an alternative higher dimensional calculus of real functions, a double calculus, which includes many two-variable extensions of classical results from single variable calculus, such as Rolle’s theorem, Lagrange’s mean value theorem, Cauchy’s mean value theorem, Fermat’s extremum theorem, the first derivative test, and the first and second fundamental theorems of calculus. 

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  • 11.
    Lundström, Patrik
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Elementär optimeringslära2021Book (Other academic)
    Abstract [sv]

    Elementär optimeringslära inleds med en repetition av grundläggande matematikkunskaper om algebra, ekvationer, matriser, funktioner och derivata. Därefter behandlas linjär optimering, först i två variabler med fokus på geometrisk förståelse och därefter, i det allmänna fallet, med simplexmetoden. Denna metod tillämpas i boken även inom det spännande matematikområdet spelteori. Boken redogör för olika metoder för optimering av allmänna tvåvariabelfunktioner, dels över kompakta områden, dels med Lagranges sats givet ett bivillkor och dels med Hessianen. Boken tar avslutningsvis upp optimering på grafer med Kruskals, Prims och Dijkstras algoritmer.

  • 12.
    Lundström, Patrik
    University West, Department of Technology, Mathematics and Computer Science.
    Fuzzified categories of composition graphs2005Report (Other (popular science, discussion, etc.))
    Abstract [en]

    Serving as a generalization of many examples of fuzzy algebraical systems equipped with a binary operation, we introduce fuzzy composition graphs and show that categories formed by such graphs are, in the sense of Wyler [10], top categories. By using this, we investigate projective and injective objects in such categories, and we determine when various limits and colimits, such as terminal and initial objects, products, coproducts, pullbacks, pushouts, equalizers, coequalizers, kernels and cokernels, exist in categories of this type and what they look like. These results are then applied to the categories of fuzzy sets, fuzzy categories, fuzzy groupoids, fuzzy monoids, fuzzy groups and fuzzy abelian groups.

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  • 13.
    Lundström, Patrik
    University West, Department of Technology. University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Generalized Brauer Algebras2002In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 30, no 5, p. 2229-2270Article in journal (Refereed)
    Abstract [en]

    Using some ideas of Brauer, we introduce what we call generalized Brauer algebras and, as a special case, Brauer orders. We show that many well-known classes of so-called crossed product algebras, and in particular, the well-known crossed product orders, can be obtained as special instances of our construction. We prove several results showing when Brauer orders are Azumaya, maximal, hereditary or Gorenstein.

  • 14.
    Lundström, Patrik
    University West, Department of Engineering Science, Divison of Natural Sciences, Surveying and Mechanical Engineering.
    Good Magma Gradings On Rings2014In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 42, no 12, p. 5357-5373Article in journal (Refereed)
    Abstract [en]

    Suppose that G and H are magmas and that R is a strongly G-graded ring. We show that there is a bijection between the set of good (zero) H-gradings of R and the set of (zero) magma homomorphisms from G to H. Thereby we generalize a result by Dascalescu, Ion, Nastasescu and Rios Montes from group gradings of matrix rings to strongly magma graded rings. We also show that there is an isomorphism between the preordered set of good (zero) H-filters on R and the preordered set of (zero) submagmas of G \times H. These results are applied to category graded rings and, in particular, to the case when G and H are groupoids. In the latter case, we use this bijection to determine the cardinality of the set of good H-gradings on R.

  • 15.
    Lundström, Patrik
    University West, Department of Engineering Science, Division of Land Surveying and Mathematics.
    Hilbert 90 for algebras with conjugation2012In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 15, no 1, p. 119-135Article in journal (Refereed)
    Abstract [en]

    We show a version of Hilbert 90 that is valid for a large class ofalgebras many of which are not commutative, distributive orassociative. This class contains the n:th iteration of theConway-Smith doubling procedure. We use our version of Hilbert 90 toparametrize all solutions in ordered fields to the norm one equation for such algebras.

  • 16.
    Lundström, Patrik
    University West, Department of Technology.
    Normal Bases for Infinite Galois Ring Extensions1999In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 79, p. 235-240Article in journal (Refereed)
  • 17.
    Lundström, Patrik
    University West, Department of Technology. University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Normal Integral Bases for Infinite Abelian Extensions2001In: Acta Arithmetica, ISSN 0065-1036, E-ISSN 1730-6264, Vol. 100, p. 79-83Article in journal (Refereed)
  • 18.
    Lundström, Patrik
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Primitives of continuous functions via polynomials2022In: International Journal of Mathematical Education in Science and Technology, ISSN 0020-739X, E-ISSN 1464-5211, p. 1-5Article in journal (Refereed)
    Abstract [en]

    In standard books on calculus the existence of primitive functions of continuous functions is proved, in one way or another, using Riemann sums. In this note we present a completely different self-contained, however probably folkloristic, proof of this existence. Our proof combines, on the one hand, the so-called Stone Weierstrass theorem on uniform approximation of continuous functions on the unit interval by polynomials, and, on the other hand, a classical result from calculus on the existence of limits of differentiated sequences of functions. The sought for primitive is then constructed as the limit of primitives of the polynomials approximating the original function.

  • 19.
    Lundström, Patrik
    University West, Department of Engineering Science, Division of Land Surveying and Mathematics.
    Pythagoreiska tripplar på sex olika sätt2008In: Normat, ISSN 0801-3500, Vol. 56, no 3, p. 111-119Article in journal (Refereed)
    Abstract [en]

    Six different ways of parametrising Pythagorean triplets are presented, from the elementary arguments supplied by the Greek, via trigonometry to Gaussian integers and applications of Hilbers 90th theorem.

  • 20.
    Lundström, Patrik
    University West, Department of Technology.
    Self-dual Normal Bases for Infinite Galois Field Extensions1998In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 26, p. 4331-4341Article in journal (Refereed)
  • 21.
    Lundström, Patrik
    University West, Department of Technology, Mathematics and Computer Science.
    Self-dual normal bases for infinite odd abelian Galois ring extensions2006In: Acta Arithmetica, ISSN 0065-1036, E-ISSN 1730-6264, Vol. 123, no 1, p. 1-8Article in journal (Refereed)
  • 22.
    Lundström, Patrik
    University West, Department of Technology, Mathematics and Computer Science.
    Self-dual Normal Integral Bases for Infinite Unramified Extensions2002In: Journal of Number Theory, ISSN 0022-314X, E-ISSN 1096-1658, Vol. 97, no 2, p. 350-367Article in journal (Refereed)
    Abstract [en]

    We prove a generalization to infinite Galois extensions of local fields, of a classical result by Noether on the existence of normal integral bases for finite tamely ramified Galois extensions. We also prove a self-dual normal integral basis theorem for infinite unramified Galois field extensions of local fields with finite residue fields of characteristic different from 2. This generalizes a result by Fainsilber for the finite case. To do this, we obtain an injectivity result concerning pointed cohomology sets, defined by inverse limits of norm-one groups of free finite-dimensional algebras with involution over complete discrete valuation rings.

  • 23.
    Lundström, Patrik
    University West, Department of Technology, Mathematics and Computer Science, Division for Mathematics and Sciences.
    Separable Groupoid Rings2006In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 34, no 8, p. 3029-3041Article in journal (Refereed)
    Abstract [en]

    We show that groupoid rings are separable over their ring of coefficients if and only if the groupoid is finite and the orders of the associated principal groups are invertible in the ring of coefficients. We use this to show that if we are given a finite groupoid, then the associated groupoid ring is semisimple (or hereditary) if and only if the ring of coefficients is semisimple (or hereditary) and the orders of the principal groups are invertible in the ring of coefficients. To this end, we extend parts of the theory of graded rings and modules from the group graded case to the category graded, and, hence, groupoid graded situation. In particular, we show that strongly groupoid graded rings are separable over their principal components if and only if the image of the trace map contains the identity

  • 24.
    Lundström, Patrik
    University West, Department of Technology, Mathematics and Computer Science.
    Separable groupoid rings2006In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 34, p. 13p. 3029-3041Article in journal (Other (popular science, discussion, etc.))
  • 25.
    Lundström, Patrik
    University West, Department of Technology, Mathematics and Computer Science.
    The category of groupoid graded modules2004In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 100, p. 15p. 195-211Article in journal (Refereed)
  • 26.
    Lundström, Patrik
    University West, Department of Technology, Mathematics and Computer Science.
    The Picard Groupoid and Strongly Groupoid Graded Modules2006In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 106, p. 1-13Article in journal (Refereed)
  • 27.
    Lundström, Patrik
    University West, Department of Engineering Science.
    The Picard Groupoid and Strongly Groupoid Graded Modules2006In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 106, p. 1-13Article in journal (Refereed)
  • 28.
    Lundström, Patrik
    University West, Department of Technology, Mathematics and Computer Science.
    Von Neumann finite endomorphism rings2003In: Indagationes mathematicae, ISSN 0019-3577, E-ISSN 1872-6100, Vol. 14, p. 9p. 223-232Article in journal (Refereed)
  • 29.
    Lundström, Patrik
    University West, Department of Engineering Science, Division of Land Surveying and Mathematics.
    Weak topological functors2008In: Journal of Generalized Lie Theory and Applications, ISSN 1736-5279, E-ISSN 1736-4337, Vol. 2, no 3, p. 211-215Article in journal (Refereed)
    Abstract [en]

    We introduce weak topological functors and show that they lift and preserve weak limits and weak colimits. We also show that if then the induced functor of Wyler’s top categories and in particular to functor categories of fuzzy maps, fuzzy relations, fuzzy topological spaces and fuzzy measurable spaces. A ! B is a topological functor and J is a category,AJ ! BJ is topological. These results are applied to a generalization

  • 30.
    Lundström, Patrik
    et al.
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Oinert, Johan
    Department of Mathematics and Natural Sciences, Blekinge Institute of Technology, SE-37179 Karlskrona (SWE).
    Group gradations on Leavitt path algebras (vol 19, 2050165, 2020)2023In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, article id 2492001Article in journal (Refereed)
    Abstract [en]

    The results that are stated in P. Nystedt and J. Oinert [Group gradations on Leavitt path algebras, J. Algebra Appl. 19(9) (2020) 2050165, Sec. 4] hold true, but due to an oversimplification some of the proofs are incomplete. The purpose of this note is to amend and complete the affected proofs.

  • 31.
    Lundström, Patrik
    et al.
    University West, Department of Engineering Science, Division of Land Surveying and Mathematics.
    Öinert, Johan
    Commutativity and Ideals in Category Crossed Products2010In: Proceedings of the Estonian Academy of Sciences: Physics, Mathematics, ISSN 1406-0086, E-ISSN 2228-0685, Vol. 59, no 4, p. 338-346Article in journal (Refereed)
  • 32.
    Lundström, Patrik
    et al.
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Öinert, Johan
    Department of Mathematics and Natural Sciences, Blekinge Institute of Technology, Karlskrona (SWE).
    Corrigendum: Group gradations on Leavitt path algebras2023In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829Article in journal (Other academic)
    Abstract [en]

    The results that are stated in P. Nystedt and J. Öinert [Group gradations on Leavitt path algebras, J. Algebra Appl. 19(9) (2020) 2050165, Sec. 4] hold true, but due to an oversimplification some of the proofs are incomplete. The purpose of this note is to amend and complete the affected proofs.

  • 33.
    Lundström, Patrik
    et al.
    University West, Department of Engineering Science.
    Öinert, Johan
    LTH.
    Miyashita Action in Strongly Groupoid Graded Rings2012In: International Electronic Journal of Algebra, E-ISSN 1306-6048, Vol. 11, p. 46-63Article in journal (Refereed)
    Abstract [en]

    We determine the commutant of homogeneous subrings in strongly groupoid graded rings in terms of an action on the ring induced by the grading. Thereby we generalize a classical result of Miyashita from the groupgraded case to the groupoid graded situation. In the end of the article we exemplify this result. To this end, we show, by an explicit construction,that given a finite groupoid $G$, equipped with a nonidentitymorphism t : d(t) -> c(t), there is a strongly G-graded ring R with the properties that each R_s, for s in G, is nonzero and R_t is a nonfree left R_c(t)-module.

  • 34.
    Lundström, Patrik
    et al.
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Öinert, Johan
    Department of Mathematics and Natural Sciences, Blekinge Institute of Technology, SE-37179 Karlskrona (SWE).
    Simplicity of Leavitt Path Algebras via Graded Ring Theory2023In: Bulletin of the Australian Mathematical Society, ISSN 0004-9727, E-ISSN 1755-1633, Vol. 108, no 3, p. 428-437Article in journal (Refereed)
    Abstract [en]

    Suppose that R is an associative unital ring and that E= (E-0, E-1, r, s) is a directed graph. Using results from graded ring theory, we show that the associated Leavitt path algebra L-R(E) is simple if and only if R is simple, E-0 has no nontrivial hereditary and saturated subset, and every cycle in E has an exit. We also give a complete description of the centre of a simple Leavitt path algebra.

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  • 35.
    Lundström, Patrik
    et al.
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    Öinert, Johan
    Köpenhamns Universitet.
    Skew category algebras associated with partially defined dynamical systems2012In: International Journal of Mathematics, ISSN 0129-167X, Vol. 23, no 4, p. 16-Article in journal (Refereed)
    Abstract [en]

    We introduce partially defined dynamical systems defined on a topological space. To each such system we associate a functor s from a category G to Top^op and show that it defines what we call a skew category algebra AxG. We study the connection between topological freeness of s and, on the one hand, ideal properties of AxG and, on the other hand, maximal commutativity of A in AxG. In particular, we show that if G is a groupoid and for each e in ob(G) the group of all morphisms from e to e is countable and the topological space s(e) is Tychonoff and Baire, then the following assertions are equivalent: (i) s is topologically free; (ii) A has the ideal intersection property, that is if I is a nonzero ideal of AxG, then I \cap A is not equal to zero; (iii) the ring A is a maximal abelian complex subalgebra of AxG. Thereby, we generalize a result by Svensson, Silvestrov and de Jeu from the additive group of integers to a large class of groupoids.

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  • 36.
    Lundström, Patrik
    et al.
    University West, Department of Engineering Science, Division of Land Surveying and Mathematics.
    Öinert, Johan
    LTH.
    The Ideal Intersection Property for Groupoid Graded Rings2012In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 40, no 5, p. 1860-1871Article in journal (Refereed)
    Abstract [en]

    We show that if a groupoid graded ring hasa certain nonzero ideal property, then the commutant of the center of the principal component of the ringhas the ideal intersection property, that is it intersects nontrivially every nonzero ideal of the ring. Furthermore, we show that for skew groupoid algebras withcommutative principal component, the principal componentis maximal commutative if and only if it has the ideal intersection property.

  • 37.
    Lundström, Patrik
    et al.
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Öinert, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences, Karlskrona (SWE).
    Richter, Johan
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences, Karlskrona (SWE).
    Non-Unital Ore Extensions2023In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 172, no 2, p. 217-229Article in journal (Refereed)
    Abstract [en]

    We study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings R[x; delta], under the hy-pothesis that R is s-unital and ker(delta) contains a non-zero idempotent. This result gener-alizes a result by oinert, Richter and Silvestrov from the unital setting. We also present a family of examples of simple non-unital differential polynomial rings.

  • 38.
    Nystedt, Patrik
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    A combinatorial proof of associativity of Ore extensions2013In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 313, no 23, p. 2748-2750Article in journal (Refereed)
    Abstract [en]

    We use a counting argument to show that Ore extensions are associative. 

  • 39.
    Nystedt, Patrik
    University West, Department of Engineering Science, Divison of Natural Sciences, Surveying and Mechanical Engineering.
    A Proof of the Cosine Addition Formula Using the Law of Cosines2014In: Mathematics Magazine, ISSN 0025-570X, E-ISSN 1930-0980, Vol. 87, no 2, p. 144-144Article in journal (Refereed)
    Abstract [en]

    We give a proof of the cosine addition formula using the law of cosines.

  • 40.
    Nystedt, Patrik
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    A proof of the law of sines using the law of cosines2017In: Mathematics Magazine, ISSN 0025-570X, E-ISSN 1930-0980, Vol. 90, no 3, p. 180-181Article in journal (Refereed)
    Abstract [en]

    We give a proof of the law of sines using the law of cosines. © Mathematical Association of America.

  • 41.
    Nystedt, Patrik
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Arc length of function graphs via Taylor’s formula2021In: International Journal of Mathematical Education in Science and Technology, ISSN 0020-739X, E-ISSN 1464-5211, Vol. 52, no 2, p. 310-323Article in journal (Refereed)
    Abstract [en]

    We use Taylor’s formula with Lagrange remainder to prove that functions with bounded second derivative are rectifiable in the case when polygonal paths are defined by interval subdivisions which are equally spaced. As a means for generating interesting examples of exact arc length calculations in calculus courses, we recall two large classes of functions f with the property that (Formula presented.) has a primitive, including classical examples by Neile, van Heuraet and Fermat, as well as more recent ones induced by Pythagorean triples of functions. We also discuss potential benefits for our proposed definition of arc length in introductory calculus courses. © 2020, © 2020 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.

  • 42.
    Nystedt, Patrik
    University West, Department of Engineering Science, Division of Mechanical Engineering and Natural Sciences.
    Fuzzy crossed product algebras2015In: Annals of Fuzzy Mathematics and Informatics, ISSN 2093-9310, E-ISSN 2287-6235, Vol. 10, no 6, p. 959-969Article in journal (Refereed)
    Abstract [en]

    We introduce fuzzy groupoid graded rings and, as a par-ticular case, fuzzy crossed product algebras. We show that there is abijection between the set of fuzzy graded is omorphism equivalence classes of fuzzy crossed product algebras and the associated second cohomology group. This generalizes a classical result for crossed product algebras to thefuzzy situation. Thereby, we quantize the difference of richness between the fuzzy and the crisp case. We give several examples showing that in the fuzzy case the associated second cohomology group is much ner than in the classical situation. In particular, we show that the cohomology group may by in nite in the fuzzy case even though it is trivial in the crisp case.

  • 43.
    Nystedt, Patrik
    University West, Department of Engineering Science.
    Limits of Fuzzy Categories2012In: Advances in Fuzzy Systems, ISSN 1687-7101, E-ISSN 1687-711X, Vol. 13, no 2, p. 77-96Article in journal (Refereed)
    Abstract [en]

    Serving as a generalization of many examples of fuzzy algebraical systems, we introduce fuzzy categories and show that categories formed by fuzzy categories are topological. By using this, we show results concerning the existence of limits and colimits in such categories. We apply these results to the categories of fuzzy sets, fuzzy categories, fuzzy groupoids, fuzzy monoids, fuzzy groups, fuzzy abelian groups and fuzzy ordered sets. Thereafter, we determine the complete ordered lattice structure of the collection of grade maps on some finite categories, in particular on cyclic groups of prime power order. We use this in the end of the article to construct grade maps on p-adic groups.

  • 44.
    Nystedt, Patrik
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Noncommutatively Graded Algebras2020In: Algebraic Structures and Applications / [ed] Silvestrov S., Malyarenko A., Rančić M. (eds), Springer, Cham: Springer International Publishing , 2020, Vol. 317, p. 371-383Conference paper (Refereed)
    Abstract [en]

    Inspired by the commutator and anticommutator algebras derived from algebras graded by groups, we introduce noncommutatively graded algebras. We generalize various classical graded results to the noncommutatively graded situation concerning identity elements, inverses, existence of limits and colimits and adjointness of certain functors. In the particular instance of noncommutatively graded Lie algebras, we establish the existence of universal graded enveloping algebras and we show a graded version of the Poincaré-Birkhoff-Witt theorem.

  • 45.
    Nystedt, Patrik
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Partial category actions on sets and topological spaces2018In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 46, no 2, p. 671-683Article in journal (Refereed)
    Abstract [en]

    We introduce (continuous) partial category actions on sets (topological spaces) and show that each such action admits a universal globalization. Thereby, we obtain a simultaneous generalization of corresponding results for groups, by Abadie, and Kellendonk and Lawson, and for monoids, by Megrelishvili and Schroder. We apply this result to the special case of partial groupoid actions where we obtain a sharpening of a result by Gilbert, concerning ordered groupoids, in the sense that mediating functions between universal globalizations always are injective.

  • 46.
    Nystedt, Patrik
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Poisson’s fundamental theorem of calculus via Taylor’s formula2020In: International Journal of Mathematical Education in Science and Technology, ISSN 0020-739X, E-ISSN 1464-5211, Vol. 51, no 5, p. 799-805Article in journal (Refereed)
    Abstract [en]

    We use Taylor’s formula with Lagrange remainder to make a modern adaptation of Poisson’s proof of a version of the fundamental theorem of calculus in the case when the integral is defined by Euler sums, that is Riemann sums with left endpoints which are equally spaced. We discuss potential benefits for such an approach in basic calculus courses. © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group.

  • 47.
    Nystedt, Patrik
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Simplicity of algebras via epsilon-strong systems2020In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 162, no 2, p. 279-301Article in journal (Refereed)
    Abstract [en]

    We obtain sufficient criteria for simplicity of systems, that is, rings R that are equipped with a family of additive subgroups R-s for s is an element of S, where S is a semigroup satisfying R = Sigma (s is an element of S) R-s and RsRt subset of R-st for s, t is an element of S. These criteria are specialized to obtain sufficient criteria for simplicity of what we call s-unital epsilon-strong systems, that is, systems where S is an inverse semigroup, R is coherent, in the sense that R-s subset of R-t for all s, t is an element of S with s <= t and for each s is an element of S, the RsRs*-Rs*Rs -bimodule R-s is s-unital. As an application, we obtain generalizations of recent criteria for simplicity of skew inverse semigroup rings by Beuter, Goncalves, Oinert and Royer, and then for Steinberg algebras over non-commutative rings by Brown, Farthing, Sims, Steinberg, Clark and Edie-Michell.

  • 48.
    Nystedt, Patrik
    et al.
    University West, Department of Engineering Science, Division of Mechanical Engineering and Natural Sciences.
    Oinert, Johan
    Lund Univ, Ctr Math Sci, SE-22100 Lund, Sweden.
    Simple semigroup graded rings2015In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 14, no 7Article in journal (Refereed)
    Abstract [en]

    We show that if R is a, not necessarily unital, ring graded by a semigroup G equipped with an idempotent e such that G is cancellative at e, the nonzero elements of eGe form a hypercentral group and R-e has a nonzero idempotent f, then R is simple if and only if it is graded simple and the center of the corner subring fR(eGe)f is a field. This is a generalization of a result of Jespers’ on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity of a, not necessarily unital, partial skew group ring by a hypercentral group. Thereby, we generalize a very recent result of Goncalves’. We also point out how Jespers’ result immediately implies a generalization of a simplicity result, recently obtained by Baraviera, Cortes and Soares, for crossed products by twisted partial actions.

  • 49.
    Nystedt, Patrik
    et al.
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    Öinert, J.
    Lund University, Centre for Mathematical Sciences.
    Simple rings and degree maps2014In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 401, p. 201-219Article in journal (Refereed)
    Abstract [en]

    For an extension A/B of neither necessarily associative nor necessarily unital rings, we investigate the connection between simplicity of A with a property that we call A-simplicity of B. By this we mean that there is no non-trivial ideal I of B being A-invariant, that is satisfying A I ⊆ I A. We show that A-simplicity of B is a necessary condition for simplicity of A for a large class of ring extensions when B is a direct summand of A. To obtain sufficient conditions for simplicity of A, we introduce the concept of a degree map for A/B. By this we mean a map d from A to the set of non-negative integers satisfying the following two conditions: (d1) if a ∈ A, then d(a) = 0 if and only if a = 0; (d2) there is a subset X of B generating B as a ring such that for each non-zero ideal I of A and each non-zero a ∈ I there is a non-zero a ' ∈ I with d(a ') ≤ d(a) and d(a 'b - ba ') &lt; d(a) for all b ∈ X. We show that if the centralizer C of B in A is an A-simple ring, every intersection of C with an ideal of A is A-invariant, A C A = A and there is a degree map for A/B, then A is simple. We apply these results to various types of graded and filtered rings, such as skew group rings, Ore extensions and Cayley-Dickson doublings. © 2013 Elsevier Inc.

  • 50.
    Nystedt, Patrik
    et al.
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Öinert, Johan
    Blekinge Institute of Technology, Department of Mathematics and Natural Sciences, Karlskrona, SE-37179, Sweden.
    Group gradations on Leavitt path algebras2020In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 19, no 9, article id 2050165Article in journal (Refereed)
    Abstract [en]

    Given a directed graph E and an associative unital ring R one may define the Leavitt path algebra with coefficients in R, denoted by LR(E). For an arbitrary group G, LR(E) can be viewed as a G-graded ring. In this paper, we show that LR(E) is always nearly epsilon-strongly G-graded. We also show that if E is finite, then LR(E) is epsilon-strongly G-graded. We present a new proof of Hazrat’s characterization of strongly g-graded Leavitt path algebras, when E is finite. Moreover, if E is row-finite and has no source, then we show that LR(E) is strongly-graded if and only if E has no sink. We also use a result concerning Frobenius epsilon-strongly G-graded rings, where G is finite, to obtain criteria which ensure that LR(E) is Frobenius over its identity component. © 2020 World Scientific Publishing Company.

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