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Mathematical Communications
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  This is an Open Access Journal Open Access journal
ISSN (Print) 1331-0623 - ISSN (Online) 1848-8013
Published by Josip Juraj Strossmayer University of Osijek Homepage  [3 journals]
  • On the power values of the sum of three squares in arithmetic progression

    • Authors: Maohua Le, Gokhan Soydan
      Pages: 137 - 150
      Abstract: In this paper, using a deep result on the existence of primitive divisors of Lehmer numbers due to Y. Bilu, G. Hanrot and P. M. Voutier, we first give an explicit formula for all positive integer solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ (*) when $n$ is an odd prime and $d=p^r$, $p>3$ a prime. So this improves the results on the papers of A. Koutsianas and V. Patel \cite{KP} and A. Koutsianas \cite{Kou}. Secondly, under the assumption of our first result, we prove that (*) has at most one solution $(x,y)$. Next, for a general $d$, we prove the following two results: (i) if every odd prime divisor $q$ of $d$ satisfies $q\not\equiv \pm 1 \pmod{2n},$ then (*) has only the solution $(x,y,d,n)=(21,11,2,3)$. (ii) if $n>228000$ and $d>8\sqrt{2}$, then all solutions $(x,y)$ of (*) satisfy $y^n<2^{3/2}d^3$.
      PubDate: 2022-10-14
      Issue No: Vol. 27, No. 2 (2022)
       
  • New regular two-graphs on 38 and 42 vertices

    • Authors: Marija Maksimović, Sanja Rukavina
      Pages: 151 - 161
      Abstract: All regular two-graphs having up to 36 vertices are known, and the first open case is the enumeration of two-graphs on 38 vertices. It is known that there are at least 191 regular two-graphs on 38 vertices and at least 18 regular two-graphs on 42 vertices. The number of descendants of these two-graphs is 6760 and 120, respectively.In this paper, we classify strongly regular graphs with parameters (41,20,9,10) having nontrivial automorphisms and show that there are exactly 7152 such graphs. We enumerate all regular two-graphs on 38 and 42 vertices with at least one descendant whose full automorphism group is nontrivial and establish that there are at least 194 regular two- graphs on 38 vertices and at least 752 regular two-graphs on 42 vertices. Furthermore, we construct descendants with trivial automorphism group of newly constructed two-graphs and increase the number of known strongly regular graphs with parameters (37,18,8,9) and (41,20,9,10) to 6802 and 18439 respectively. This significantly increases the number of known strongly regular graphs with parameters (41,20,9,10).
      PubDate: 2022-10-14
      Issue No: Vol. 27, No. 2 (2022)
       
  • Complete solution of the exponential Diophantine equation
           $P_n^x+P_{n+1}^x=P_m^y$

    • Authors: Bernadette Faye, Carlos Gomez, Florian Luca, Salah Rihane, Alain Togbe
      Pages: 163 - 185
      Abstract: In this paper, we find all the solutions of the title Diophantine equation in positive integer variables $(m, n, x,y)$, where $P_k$ is the $k$th term of the Pell sequence.
      PubDate: 2022-10-14
      Issue No: Vol. 27, No. 2 (2022)
       
  • A sensitivity study of the artificial viscosity in defect deferred
           correction method for the coupled Stokes/Darcy model

    • Authors: Yanan Yang, Pengzhan Huang
      Pages: 187 - 202
      Abstract: This paper analyzes the sensitivity of the artificial viscosity in the defect deferred correction method for the non-stationary coupled Stokes/Darcy model. For the defect step and the deferred correction step of the defect deferred correction method, we respectively give the corresponding sensitivity systems related to the change of artificial viscosity. Finite element schemes are devised for computing solutions to the sensitivity systems. Finally, we will verify the theoretical analysis results through numerical experiments. Our results reveal that the solution is sensitive for small values of the artificial viscosity, and when the viscosity/hydraulic conductivity coefficients are small.
      PubDate: 2022-10-14
      Issue No: Vol. 27, No. 2 (2022)
       
  • Alternating Sums of Binomial Quotients

    • Authors: Wenchang Chu, Dongwei Guo
      Pages: 203 - 212
      Abstract: By combining telescoping and the linearization method,
      a class of alternating sums of binomial quotients are
      investigated. Several summation and transformation
      formulae are established. Asymptotic behavior
      for these sums is also examined.
      PubDate: 2022-10-14
      Issue No: Vol. 27, No. 2 (2022)
       
  • A note on the Fibonacci $m$-step sequences modulo $q$

    • Authors: Karol Gryszka
      Pages: 213 - 221
      Abstract: We briefly discuss a congruence relation on the subsequence of the Fibonacci $m$-step numbers. Then we use the obtained result to establish many identities regarding the Tribonacci numbers mod $q$ with indices in arithmetic progression and connect them with the existing results. Finally, we discuss arbitrary linear recurrence sequences.
      PubDate: 2022-10-14
      Issue No: Vol. 27, No. 2 (2022)
       
  • An optimality property of approximated solution computed by Hessenberg
           method

    • Authors: Mehdi Najafi-Kalyani, Fatemeh P. A. Beik
      Pages: 223 - 237
      Abstract: We  revisit the implementation of Krylov subspace method based on the Hessenberg process for solving general linear operator equations. More precisely, it is established that the computed approximate solution by corresponding approach  can be regarded as the minimizer of a certain norm of system's residual at each step.  Test problems are numerically examined to compare the performance of Hessenberg method with Krylov subspace method based on the Arnoldi process in conjunction with the Tikhonov regularization technique for solving tensor equations with cosine transform product arising from image restoration.
      PubDate: 2022-10-14
      Issue No: Vol. 27, No. 2 (2022)
       
  • Periodic Solutions For Fourth-order Differential Equation Set On Singular
           Domain In R N+1

    • Authors: Chaouchi Belkacem, Marko Kostić, Daniel Velinov
      Pages: 239 - 253
      Abstract: The present work is devoted to the study of a fourt-order differential equation set on a singular domain in R N+1. The existence and uniqueness of a periodic strict solution are discussed in the framework of H¨older spaces
      PubDate: 2022-10-14
      Issue No: Vol. 27, No. 2 (2022)
       
  • Existence and Uniqueness of a Periodic Solution to a Certain Third-Order
           Neutral Functional Differential Equation

    • Authors: Rasha Osman Ahmed Taie, Mona Ghaleb Abdullah Alwaleedy
      Pages: 255 - 274
      Abstract: In this paper, by applying Mawhin's continuation theorem of the coincidence
      degree theory, some sufficient conditions for the existence and uniqueness of an $\omega$-periodic solution for the following third-order neutral functional differential equation are established\dfrac{d^{3}}{dt^{3}}\bigg ( x(t)-d(t)x\big (t-\delta(t)\big ) \bigg )+a(t)\ddot{x}(t)+b(t)f\big (t,\dot{x}(t)\big )+\sum_{i=1}^{n}c_{i}(t)g\big (t,x(t-\tau_{i}(t))\big )=e(t).Moreover, we present an example and a graph to demonstrate the validity of analytical conclusion.
      PubDate: 2022-10-14
      Issue No: Vol. 27, No. 2 (2022)
       
  • A limit formula for real Richardson orbits

    • Authors: Mladen Božičević
      Pages: 275 - 286
      Abstract: Let $G_\mathbb R$ be a real, semisimple, linear and connected Lie group. Let $K$ denote the complexification of a maximal compact group of $G_\mathbb R$. Assume that $G_\mathbb R$
      has a compact Cartan subgroup. We prove a formula which computes the Liouville measure on a real nilpotent Richardson orbit,
      obtained by the Sekiguchi correspondence from a $K$-nilpotent Richardson orbit, as a limit of differentiated measures on regular elliptic orbits.
      PubDate: 2022-10-14
      Issue No: Vol. 27, No. 2 (2022)
       
 
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