Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 9Citation - Scopus: 8A Numerical Method Based on Legendre Wavelet and Quasilinearization Technique for Fractional Lane-Emden Type Equations(Springer, 2024) İdiz, F.; Tanoǧlu, G.; Aghazadeh, N.In this research, we study the numerical solution of fractional Lane-Emden type equations, which emerge mainly in astrophysics applications. We propose a numerical approach making use of Legendre wavelets and the quasilinearization technique. The nonlinear term in fractional Lane-Emden type equations is iteratively linearized using the quasilinearization technique. The linearized equations are then solved using the Legendre wavelet collocation method. The proposed method is quite effective to overcome the singularity in fractional Lane-Emden type equations. Convergence and error analysis of the proposed method are given. We solve some test problems to compare the effectiveness of the proposed method with some other numerical methods in the literature. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.Conference Object Citation - Scopus: 2Maximally Entangled Two-Qutrit Quantum Information States and De Gua’s Theorem for Tetrahedron(Springer, 2023) Pashaev, OktayGeometric relations between separable and entangled two-qubit and two-qutrit quantum information states are studied. For two qubit states a relation between reduced density matrix and the concurrence allows us to characterize entanglement by double area of a parallelogram, expressed by determinant of the complex Hermitian inner product metric. We find similar relation in the case of generic two-qutrit state, where the concurrence is expressed by sum of all 2 × 2 minors of 3 × 3 complex matrix. We show that for maximally entangled two-retrit state this relation is just De Gua’s theorem or a three-dimensional analog of the Pythagorean theorem for triorthogonal tetrahedron areas. Generalizations of our results for arbitrary two-qudit states are discussed © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.Conference Object Hirota Bilinear Method and Relativistic Dissipative Soliton Solutions in Nonlinear Spinor Equations(Springer, 2023) Pashaev, OktayA new relativistic integrable nonlinear model for real, Majorana type spinor fields in 1+1 dimensions, gauge equivalent to Papanicolau spin model, defined on the one sheet hyperboloid is introduced. By using the double numbers, the model is represented as hyperbolic complex valued relativistic massive Thirring type model. By Hirota’s bilinear method, an exact one and two dissipative soliton solutions of this model are constructed. Calculation of first three integrals of motion for one dissipation solution shows that the last one represents a particle-like nonlinear excitation, with relativistic dispersion and highly nonlinear mass. A nontrivial solution of the system of algebraic equations, showing fusion and fission of relativistic dissipations is found. Asymptotic analysis of exact two dissipaton solution confirms resonant character of our dissipaton interactions. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.Conference Object Citation - Scopus: 1Pq-Calculus of Fibonacci Divisors and Method of Images in Planar Hydrodynamics(Springer, 2022) Pashaev, OktayBy introducing the hierarchy of Fibonacci divisors and corresponding quantum derivatives, we develop the golden calculus, hierarchy of golden binomials and related exponential functions, translation operator and infinite hierarchy of Golden analytic functions. The hierarchy of Golden periodic functions, appearing in this calculus we relate with the method of images in planar hydrodynamics for incompressible and irrotational flow in bounded domain. We show that the even hierarchy of these functions determines the flow in the annular domain, bounded by concentric circles with the ratio of radiuses in powers of the Golden ratio. As an example, complex potential and velocity field for the set of point vortices with Golden proportion of images are calculated explicitly.Conference Object Citation - Scopus: 5Kaleidoscope of Classical Vortex Images and Quantum Coherent States(Springer, 2018) Pashaev, Oktay; Koçak, AygülThe Schrödinger cat states, constructed from Glauber coherent states and applied for description of qubits are generalized to the kaleidoscope of coherent states, related with regular n-polygon symmetry and the roots of unity. This quantum kaleidoscope is motivated by our method of classical hydrodynamics images in a wedge domain, described by q-calculus of analytic functions with q as a primitive root of unity. First we treat in detail the trinity states and the quartet states as descriptive for qutrit and ququat units of quantum information. Normalization formula for these states requires introduction of specific combinations of exponential functions with mod 3 and mod 4 symmetry, which are known also as generalized hyperbolic functions. We show that these states can be generated for an arbitrary n by the Quantum Fourier transform and can provide in general, qudit unit of quantum information. Relations of our states with quantum groups and quantum calculus are discussed. © Springer Nature Switzerland AG 2018.Conference Object Citation - WoS: 25Citation - Scopus: 26Quantum Key Distribution in the Classical Authenticated Key Exchange Framework(Springer, 2013) Mosca, Michele; Stebila, Douglas; Ustaoğlu, BerkantKey establishment is a crucial primitive for building secure channels in a multi-party setting. Without quantum mechanics, key establishment can only be done under the assumption that some computational problem is hard. Since digital communication can be easily eavesdropped and recorded, it is important to consider the secrecy of information anticipating future algorithmic and computational discoveries which could break the secrecy of past keys, violating the secrecy of the confidential channel. Quantum key distribution (QKD) can be used generate secret keys that are secure against any future algorithmic or computational improvements. QKD protocols still require authentication of classical communication, although existing security proofs of QKD typically assume idealized authentication. It is generally considered folklore that QKD when used with computationally secure authentication is still secure against an unbounded adversary, provided the adversary did not break the authentication during the run of the protocol. We describe a security model for quantum key distribution extending classical authenticated key exchange (AKE) security models. Using our model, we characterize the long-term security of the BB84 QKD protocol with computationally secure authentication against an eventually unbounded adversary. By basing our model on traditional AKE models, we can more readily compare the relative merits of various forms of QKD and existing classical AKE protocols. This comparison illustrates in which types of adversarial environments different quantum and classical key agreement protocols can be secure. © 2013 Springer-Verlag.