Blar i forfatter "Høibakk, Ralph"

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    • Geometric Construction of Some Lehmer Means 

      Høibakk, Ralph; Lukkassen, Dag; Meidell, Annette; Persson, Lars Erik (Journal article; Tidsskriftartikkel; Peer reviewed, 2018-11-14)
      The main aim of this paper is to contribute to the recently initiated research concerning geometric constructions of means, where the variables are appearing as line segments. The present study shows that all Lehmer means of two variables for integer power k and for k = m 2 , where m is an integer, can be geometrically constructed, that Lehmer means for power k = 0,1 and 2 can be geometrically ...

    • A New Development of the Classical Single Ladder Problem via Converting the Ladder to a Staircase 

      Høibakk, Ralph; Lukkassen, Dag; Meidell, Annette; Persson, Lars Erik (Journal article; Tidsskriftartikkel; Peer reviewed, 2021-02-08)
      Our purpose is to shed some new light on problems arising from a study of the classical Single Ladder Problem (SLP). The basic idea is to convert the SLP to a corresponding Single Staircase Problem. The main result (Theorem 1) shows that this idea works fine and new results can be obtained by just calculating rational solutions of an algebraic equation. Some examples of such concrete calculations ...

    • A New Look at the Single Ladder Problem (SLP) via Integer Parametric Solutions to the Corresponding Quartic Equation 

      Høibakk, Ralph; Lukkassen, Dag; Meidell, Annette; Persson, Lars Erik (Journal article; Tidsskriftartikkel; Peer reviewed, 2020-02-18)
      The aim is to put new light on the single ladder problem (SLP). Some new methods for finding complete integer solutions to the corresponding quartic equation z 4 −2L z 3 +( L 2 − a 2 − b 2 ) z 2 +2L a 2 z− L 2 a 2 =0 z4−2Lz3+(L2−a2−b2)z2+2La2z−L2a2=0 are developed. For the case L≥ L min L≥Lmin , these methods imply a complete parametric representation for integer ...

    • On geometric construction of some power means 

      Høibakk, Ralph; Lukkassen, Dag; Persson, Lars Erik; Meidell, Annette (Journal article; Peer reviewed, 2018-11-27)
      In the homogenization theory, there are many examples where the effective conductivities of composite structures are power means of the local conductivities. The main aim of this paper is to initiate research concerning geometric construction of some power means of three or more variables. We contribute by giving methods for the geometric construction of the harmonic mean $ P_{-1} $ and the arithmetic ...

    • On Some Power Means and Their Geometric Constructions 

      Høibakk, Ralph; Lukkassen, Dag; Meidell, Annette; Persson, Lars Erik (Journal article; Peer reviewed; Tidsskriftartikkel, 2018)
      The main aim of this paper is to further develop the recently initiatedresearch concerning geometric construction of some power means wherethe variables are appearing as line segments. It will be demonstratedthat the arithmetic mean, the harmonic mean and the quadratic meancan be constructed for any number of variables and that all power meanswhere the number of variables are n = 2m, m 1 2 N for all ...

    • Solutions to some problems related to Diophantine equation, power means and homogenization theory 

      Høibakk, Ralph (Doctoral thesis; Doktorgradsavhandling, 2017-05-19)
      This Ph.D. thesis consists of an introduction and 7 papers where we investigate the requirements for finding integer or rational solutions to a selection of Diophantine equations leading to problems connected to power means and homogenization. In Paper 1 we present a modern view of classic number theory in a historic context. In Paper 2 we introduce the Crossed Ladders Problem and present a proof ...