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    Spectrum of a q-deformed Schrödinger equation by means of the variational method
    (Mathematical Methods in the Applied Sciences, 2023-12) Doğan Çalışır, Ayşe ; Turan, Mehmet ; Sevinik Adıgüzel, Rezan
    In this work, the q-deformed Schrödinger equations defined in different form of the q - Hamil tonian for q-harmonic oscillator studied in [5] are considered with symmetric, asymmetric and non-polynomial potentials. The spectrum of the q-Hamiltonian is obtained by using the Rayleigh Ritz variational method in which the discrete q-Hermite I polynomials are taken as the basis. As applications, q-harmonic, purely q-quartic and q-quartic oscillators are examined in the class of symmetric polynomial potentials. Moreover, the q-version of Gaussian potential for an example of a non-polynomial symmetric potential and a specific example of q-version of asymmetric dou ble well potential are presented. Numerous results are given for these potentials for several values of q. The limit relation as q → 1 − is discussed. The obtained results of ground- and excited-state energies of the purely q-quartic oscillator and the accuracy of the ground-state energy levels are compared with the existing results. Also, the results are compared with the classical case appearing in the literature in the limiting case q → 1 −.
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    Global energy preserving model reduction for multi-symplectic PDEs
    (Applied Mathematics and Computation, 2023-01-01) Uzunca, Murat ; Karasözen, Bülent ; Aydın, Ayhan
    Many Hamiltonian systems can be recast in multi-symplectic form. We develop a reduced-order model (ROM) for multi-symplectic Hamiltonian partial differential equations (PDEs) that preserves the global energy. The full-order solutions are obtained by finite difference discretization in space and the global energy preserving average vector field (AVF) method. The ROM is constructed in the same way as the full-order model (FOM) applying proper orthogonal decomposition (POD) with the Galerkin projection. The reduced-order system has the same structure as the FOM, and preserves the discrete reduced global energy. Applying the discrete empirical interpolation method (DEIM), the reduced-order solutions are computed efficiently in the online stage. A priori error bound is derived for the DEIM approximation to the nonlinear Hamiltonian. The accuracy and computational efficiency of the ROMs are demonstrated for the Korteweg de Vries (KdV) equation, Zakharov-Kuznetzov (ZK) equation, and nonlinear Schrödinger (NLS) equation in multi-symplectic form. Preservation of the reduced energies shows that the reduced-order solutions ensure the long-term stability of the solutions.
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    On the image of the Lupaş q-analogue of the Bernstein Operators
    (Bulletin of the Malaysian Mathematical Sciences Society, 2024-01) Gürel Yılmaz, Övgü ; Ostrovska, Sofiya ; Turan, Mehmet
    The Lupa\c{s} $q$-analogue, $R_{n,q}$, is historically the first known $q$-version of the Bernstein operator. It has been studied extensively in different aspects by a number of authors during the last decades. In this work, the following issues related to the image of the Lupa\c{s} $q$-analogue are discussed: A new explicit formula for the moments has been derived, independence of the image $R_{n,q}$ from the parameter $q$ has been examined, the diagonalizability of operator $R_{n,q}$ has been proved and the fact that $R_{n,q}$ does not preserve modulus of continuity has been established.
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    Classification of some quadrinomials over finite fields of odd characteristic
    (Finite Fields and Their Applications, 2023-03-15) Özbudak, Ferruh ; Gülmez Temür, Burcu
    In this paper, we completely determine all necessary and su cient conditions such that the polynomial f (x) = x3+axq+2+bx2q+1+cx3q , where a, b, c ∈ F∗ q , is a permutation quadrinomial of Fq2 over any nite eld of odd characteristic. This quadrinomial has been studied rst in [25] by Tu, Zeng and Helleseth, later in [24] Tu, Liu and Zeng revisited these quadrinomials and they proposed a more comprehensive characterization of the coe cients that results with new permutation quadrinomials, where char(Fq ) = 2 and nally, in [16], Li, Qu, Li and Chen proved that the su cient condition given in [24] is also necessary and thus completed the solution in even characteristic case. In [6] Gupta studied the permutation properties of the polynomial x3 +axq+2 +bx2q+1 +cx3q , where char(Fq ) = 3, 5 and a, b, c ∈ F∗q and proposed some new classes of permutation quadrinomials of Fq2 . In particular, in this paper we classify all ermutation polynomials of Fq2 of the form f (x) = x3 + axq+2 + bx2q+1 + cx3q , where a, b, c ∈ F∗q , over all nite elds of odd characteristic and obtain several new classes of such permutation quadrinomials.
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    Complete characterization of some permutation polynomials of the form xr(1 + axs1(q−1) + bxs2(q−1)) over Fq2
    (Cryptography and Communications, 2023-04-18) Özbudak, Ferruh ; Gülmez Temür, Burcu
    We completely characterize all permutation trinomials of the form f (x) = x3(1+axq−1 + bx2(q−1)) over Fq2 , where a, b ∈ F∗q and all permutation trinomials of the form f (x) = x3(1+bx2(q−1) +cx3(q−1)) over Fq2 , where b, c ∈ F∗q in both even and odd characteristic cases.