Mathematics / Matematik

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  • Article
    Arithmetic Progressions in Certain Subsets of Finite Fields
    (Elsevier, 2023) Eyidoğan, Sadık; Göral, Haydar; Kutlu, Mustafa Kutay
    In this note, we focus on how many arithmetic progressions we have in certain subsets of finite fields. For this purpose, we consider the sets Sp = {t2 : t & ISIN; Fp} and Cp = {t3 : t & ISIN; Fp}, and we use the results on Gauss and Kummer sums. We prove that for any integer k & GE; 3 and for an odd prime number p, the number of k-term arithmetic progressions in Sp is given by p2 2k + R, where and ck is a computable constant depending only on k. The proof also uses finite Fourier analysis and certain types of Weil estimates. Also, we obtain some formulas that give the exact number of arithmetic progressions of length  in the set Sp when  & ISIN; {3,4, 5} and p is an odd prime number. For  = 4, 5, our formulas are based on the number of points on
  • Conference Object
    Existence and Uniqueness of Solution for Discontinuous Conewise Linear Systems
    (Elsevier, 2020) Şahan, Gökhan
    In this study, we give necessary and sufficient conditions for well posedness of Conewise Linear Systems in 3-dimensional space where the vector field is allowed to be discontinuous. The conditions are stated using the subspaces derived from subsystem matrices and the results are compared with the existing conditions given in the literature. We show that even we don't have a fixed structure on system matrices as in bimodal systems, similar subspaces and numbers again determines well posedness. Copyright (C) 2020 The Authors.