Zafer Ünal
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Zafer Ünal has a total of 10 co-authors. Here are the top co-authors by number of shared publications:
- Tuncay Deniz Şentürk
Göl Anatolian Highschool, Kastamonu, Turkey
3 shared publications - Ahmet Daşdemir
Department of Mathematics, Faculty of Science, Kastamonu University, Kastamonu, Türkiye
2 shared publications - Can Doğan Vurdu
Kastamonu University
2 shared publications - Göksal Bilgici
Kastamonu University
2 shared publications Göksal Bi̇lGi̇cİ
1 shared publication- Turhan Koprubasi
Kastamonu University
1 shared publication
Latest Publications
- Open Access
- 12 October 2026
- Published by MDPI AG in Mathematics
- Vol. 14 (20) , 3693
Fibonacci–Lucas Recurrence-Indexed Quaternionic Unitaries: Channel Convergence, Clifford Proximity, and Controlled Entanglement
We investigate the normalized operator geometry of the mixed Fibonacci–Lucas quaternion Qn = Fn + Lni + Fn+1j + Ln+1k. Standard quaternion representations associate this sequence with single-qubit unitary channels. We first establish a convergence theorem for quaternion sequences with two distinct real characteristic roots and a strictly dominant positive root. It gives an exact average gate infidelity to the limit and sharp projective-step asymptotics, including the degenerate constant-channel case. For the Fibonacci–Lucas assignment, orthogonal Binet coefficient vectors yield exact finite-index formulas with projective steps of order φ−2n and average gate infidelities of order φ−4n. Strict comparison of the five projective Clifford rotation classes proves that the limiting unitary has a unique nearest projective Clifford, the cyclic Pauli-axis permutation, with average gate fidelity For the controlled family, product-input maximization gives concurrence while independent Haar averaging gives linear-entropy entangling power Both admit explicit algebraic limits. Reproducible matrix and high-precision scalar calculations verify these results. The analysis supplies exact convergence, finite-group, and entanglement benchmarks for a recurrence-indexed family without assuming Hamiltonian dynamics.- 12 October 2026
- Published by MDPI AG in Mathematics
- Vol. 14 (20) , 3693
Open AccessFibonacci–Lucas Recurrence-Indexed Quaternionic Unitaries: Channel Convergence, Clifford Proximity, and Controlled Entanglement
We investigate the normalized operator geometry of the mixed Fibonacci–Lucas quaternion Qn = Fn + Lni + Fn+1j + Ln+1k. Standard quaternion representations associate this sequence with single-qubit unitary channels. We first establish a convergence theorem for quaternion sequences with two distinct real characteristic roots and a strictly dominant positive root. It gives an exact average gate infidelity to the limit and sharp projective-step asymptotics, including the degenerate constant-channel case. For the Fibonacci–Lucas assignment, orthogonal Binet coefficient vectors yield exact finite-index formulas with projective steps of order φ−2n and average gate infidelities of order φ−4n. Strict comparison of the five projective Clifford rotation classes proves that the limiting unitary has a unique nearest projective Clifford, the cyclic Pauli-axis permutation, with average gate fidelity For the controlled family, product-input maximization gives concurrence while independent Haar averaging gives linear-entropy entangling power Both admit explicit algebraic limits. Reproducible matrix and high-precision scalar calculations verify these results. The analysis supplies exact convergence, finite-group, and entanglement benchmarks for a recurrence-indexed family without assuming Hamiltonian dynamics. - Open Access
- 13 July 2026
- Published by Springer Nature in Research Square
A Complex-Quaternionic Riemann–Silberstein Formalism for Single-Photon Electromagnetic Wave Packets
Single photons are elementary excitations of the quantized electromagnetic field, yet their electromagnetic structure is often described through separate electric and magnetic field operators. Here we develop a theorem-based complex-quaternionic Riemann-Silberstein representation of single-photon electromagnetic wave packets. The formalism embeds the transverse electric and magnetic components into one helicity-resolved biquaternionic field, so that transversality, electric-magnetic orthogonality, helicity projection, and mode-wise nullity are encoded within a single algebraic structure. We show that the source-free Maxwell equations reduce to one first-order quaternionic Maxwell equation and that canonical quantization yields the one-photon electromagnetic amplitude as a vacuum-to-one-photon transition matrix element. The resulting expression is a spectral superposition of transverse modes weighted by the photon wave-packet envelope. Beyond notational compactness, the construction separates three operational ingredients of a one-photon state: spectral localization, helicity selection, and mode matching to a detector or target mode. The framework is fully compatible with standard QED and clarifies that broadband photon wave packets are generally not pointwise null, although each Fourier component is null.
- 13 July 2026
- Published by Springer Nature in Research Square
Open AccessA Complex-Quaternionic Riemann–Silberstein Formalism for Single-Photon Electromagnetic Wave Packets
Single photons are elementary excitations of the quantized electromagnetic field, yet their electromagnetic structure is often described through separate electric and magnetic field operators. Here we develop a theorem-based complex-quaternionic Riemann-Silberstein representation of single-photon electromagnetic wave packets. The formalism embeds the transverse electric and magnetic components into one helicity-resolved biquaternionic field, so that transversality, electric-magnetic orthogonality, helicity projection, and mode-wise nullity are encoded within a single algebraic structure. We show that the source-free Maxwell equations reduce to one first-order quaternionic Maxwell equation and that canonical quantization yields the one-photon electromagnetic amplitude as a vacuum-to-one-photon transition matrix element. The resulting expression is a spectral superposition of transverse modes weighted by the photon wave-packet envelope. Beyond notational compactness, the construction separates three operational ingredients of a one-photon state: spectral localization, helicity selection, and mode matching to a detector or target mode. The framework is fully compatible with standard QED and clarifies that broadband photon wave packets are generally not pointwise null, although each Fourier component is null.
- 1 April 2023
- Published by Black Sea Journal of Engineering and Science in Black Sea Journal of Engineering and Science
- Vol. 6 (2) , 98-101
Vajda’s Identities for Dual Fibonacci and Dual Lucas Sedenions
Fibonacci and Lucas numbers have been the most popular integer sequences since they were defined. These integer sequences have many uses, from nature to computer science, from art to financial analysis. Many researchers have worked on this subject. Sedenions form a 16-dimensional algebra on the field of real numbers. Various systems can be constructed by using the terms of special integer sequences instead of terms in sedenions. In this study, we define dual Fibonacci (DFS) and dual Lucas sedenions (DLS) with the help of Fibonacci and Lucas termed sedenions. Then we calculate some special identities for DFS and DLS such as Vajda's, Catalan's, d'Ocagne's, Cassini's.
- 1 April 2023
- Published by Black Sea Journal of Engineering and Science in Black Sea Journal of Engineering and Science
- Vol. 6 (2) , 98-101
Vajda’s Identities for Dual Fibonacci and Dual Lucas Sedenions
Fibonacci and Lucas numbers have been the most popular integer sequences since they were defined. These integer sequences have many uses, from nature to computer science, from art to financial analysis. Many researchers have worked on this subject. Sedenions form a 16-dimensional algebra on the field of real numbers. Various systems can be constructed by using the terms of special integer sequences instead of terms in sedenions. In this study, we define dual Fibonacci (DFS) and dual Lucas sedenions (DLS) with the help of Fibonacci and Lucas termed sedenions. Then we calculate some special identities for DFS and DLS such as Vajda's, Catalan's, d'Ocagne's, Cassini's.
- 25 May 2022
- Published by Springer Nature in Computational Methods and Function Theory
- Vol. 22 (3) , 575-608
3-Parameter Generalized Quaternions
- 25 May 2022
- Published by Springer Nature in Computational Methods and Function Theory
- Vol. 22 (3) , 575-608
3-Parameter Generalized Quaternions
- 31 December 2021
- Published by Turkish Journal of Mathematics and Computer Science, Association of Mathematicians in Turkish Journal of Mathematics and Computer Science
- Vol. 13 (2) , 226-233
On Pell and Pell-Lucas Generalized Octonions
In this study, we gave a generalization on Pell and Pell-Lucas octonions over the algebra $\mathbb{O}(a,b,c)$ where $a,b$ and $c$ are real numbers. For these number sequences, we obtain Binet formulas and gave some well-known identities such as Catalan's identity, Cassini's identity and d'Ocagne's identity.- 31 December 2021
- Published by Turkish Journal of Mathematics and Computer Science, Association of Mathematicians in Turkish Journal of Mathematics and Computer Science
- Vol. 13 (2) , 226-233
On Pell and Pell-Lucas Generalized Octonions
In this study, we gave a generalization on Pell and Pell-Lucas octonions over the algebra $\mathbb{O}(a,b,c)$ where $a,b$ and $c$ are real numbers. For these number sequences, we obtain Binet formulas and gave some well-known identities such as Catalan's identity, Cassini's identity and d'Ocagne's identity.- Open Access
- 8 December 2020
- Published by Hacettepe University in Hacettepe Journal of Mathematics and Statistics
- Vol. 49 (6) , 2046-2062
On recursive hyperbolic functions in Fibonacci-Lucas sense
The hyperbolic Fibonacci and hyperbolic Lucas functions have been introduced before and have been improved to functions of the symmetrical form. In this paper, we generalize the mentioned definitions, which will be called Horadam hyperbolic sine function $ \left( HSF \right) $ and Horadam hyperbolic cosine function $ \left( HCF \right) $. Further, we present many identities and hyperbolic properties of our new definitions.- 8 December 2020
- Published by Hacettepe University in Hacettepe Journal of Mathematics and Statistics
- Vol. 49 (6) , 2046-2062
Open AccessOn recursive hyperbolic functions in Fibonacci-Lucas sense
The hyperbolic Fibonacci and hyperbolic Lucas functions have been introduced before and have been improved to functions of the symmetrical form. In this paper, we generalize the mentioned definitions, which will be called Horadam hyperbolic sine function $ \left( HSF \right) $ and Horadam hyperbolic cosine function $ \left( HCF \right) $. Further, we present many identities and hyperbolic properties of our new definitions. - 8 July 2020
- Published by The Scientific and Technological Research Council of Turkey (TUBITAK-ULAKBIM) - DIGITAL COMMONS JOURNALS in TURKISH JOURNAL OF MATHEMATICS
- Vol. 44 (4) , 1212-1221
A Study on Horadam Hybrid Numbers
- 8 July 2020
- Published by The Scientific and Technological Research Council of Turkey (TUBITAK-ULAKBIM) - DIGITAL COMMONS JOURNALS in TURKISH JOURNAL OF MATHEMATICS
- Vol. 44 (4) , 1212-1221
A Study on Horadam Hybrid Numbers
- 10 September 2016
- Published by Springer Nature in Advances in Applied Clifford Algebras
- Vol. 27 (2) , 1907-1916
Some Properties of Dual Fibonacci and Dual Lucas Octonions
- 10 September 2016
- Published by Springer Nature in Advances in Applied Clifford Algebras
- Vol. 27 (2) , 1907-1916
Some Properties of Dual Fibonacci and Dual Lucas Octonions