Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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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.Article Relativistic Dissipatons in Integrable Nonlinear Majorana Type Spinor Model(Yunnan University, 2022) Pashaev, Oktay; Lee, J. H.By method of moving frame, the relativistic integrable nonlinear model for real, Majorana type spinor fields in 1+1 dimensions is introduced and gauge equivalence of this model with Papanicolau spin model on one sheet hyperboloid is established. In terms of the so called double numbers, the model is represented also as hyperbolic complex relativistic model, in the form similar to the massive Thirring model. By using Hirota's bilinear method, the one dissipaton solution of this model is constructed. We calculate first integrals of motion for this dissipaton and show that it represents a relativistic particle with highly nonlinear mass. Analyzing resonance conditions for scattering of two relativistic dissipatons, we find a solution describing resonant property of the dissipatons.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.Article Citation - WoS: 17Citation - Scopus: 17Quantum Calculus of Fibonacci Divisors and Infinite Hierarchy of Bosonic-Fermionic Golden Quantum Oscillators(World Scientific Publishing, 2021) Pashaev, OktayStarting from divisibility problem for Fibonacci numbers, we introduce Fibonacci divisors, related hierarchy of Golden derivatives in powers of the Golden Ratio and develop corresponding quantum calculus. By this calculus, the infinite hierarchy of Golden quantum oscillators with integer spectrum determined by Fibonacci divisors, the hierarchy of Golden coherent states and related Fock-Bargman representations in space of complex analytic functions are derived. It is shown that Fibonacci divisors with even and odd kappa describe Golden deformed bosonic and fermionic quantum oscillators, correspondingly. By the set of translation operators we find the hierarchy of Golden binomials and related Golden analytic functions, conjugate to Fibonacci number F-kappa. In the limit. kappa -> 0, Golden analytic functions reduce to classical holomorphic functions and quantum calculus of Fibonacci divisors to the usual one. Several applications of the calculus to quantum deformation of bosonic and fermionic oscillator algebras, R-matrices, geometry of hydrodynamic images and quantum computations are discussed.Article Citation - WoS: 4Relativistic Burgers and Nonlinear Schrödinger Equations(Pleiades Publishing, 2009) Pashaev, OktayWe construct relativistic complex Burgers-Schrodinger and nonlinear Schrodinger equations. In the nonrelativistic limit, they reduce to the standard Burgers and nonlinear Schrodinger equations and are integrable through all orders of relativistic corrections.Conference Object Citation - WoS: 3Citation - Scopus: 4Special Functions With Mod N Symmetry and Kaleidoscope of Quantum Coherent States(IOP Publishing, 2019) Koçak, Aygül; Pashaev, OktayThe set of mod n functions associated with primitive roots of unity and discrete Fourier transform is introduced. These functions naturally appear in description of superposition of coherent states related with regular polygon, which we call kaleidoscope of quantum coherent states. Displacement operators for kaleidoscope states are obtained by mod n exponential functions with operator argument and non-commutative addition formulas. Normalization constants, average number of photons, Heinsenberg uncertainty relations and coordinate representation of wave functions with mod n symmetry are expressed in a compact form by these functions.Conference Object Citation - WoS: 1Citation - Scopus: 2Apollonius Representation and Complex Geometry of Entangled Qubit States(IOP Publishing, 2019) Parlakgörür, Tuğçe; Pashaev, OktayA representation of one qubit state by points in complex plane is proposed, such that the computational basis corresponds to two fixed points at a finite distance in the plane. These points represent common symmetric states for the set of quantum states on Apollonius circles. It is shown that, the Shannon entropy of one qubit state depends on ratio of probabilities and is a constant along Apollonius circles. For two qubit state and for three qubit state in Apollonius representation, the concurrence for entanglement and the Cayley hyperdeterminant for tritanglement correspondingly, are constant on the circles as well. Similar results are obtained also for n- tangle hyperdeterminant with even number of qubit states. It turns out that, for arbitrary multiple qubit state in Apollonius representation, fidelity between symmetric qubit states is also constant on Apollonius circles. According to these, the Apollonius circles are interpreted as integral curves for entanglement characteristics. The bipolar and the Cassini representations for qubit state are introduced, and their relations with qubit coherent states are established. We proposed the differential geometry for qubit states in Apollonius representation, defined by the metric on a surface in conformal coordinates, as square of the concurrence. The surfaces of the concurrence, as surfaces of revolution in Euclidean and Minkowski spaces are constructed. It is shown that, curves on these surfaces with constant Gaussian curvature becomes Cassini curves.Conference Object The Hirota Method for Reaction-Diffusion Equations With Three Distinct Roots(American Institute of Physics, 2004) Tanoğlu, Gamze; Pashaev, OktayThe Hirota Method, with modified background is applied to construct exact analytical solutions of nonlinear reaction-diffusion equations of two types. The first equation has only nonlinear reaction part, while the second one has in addition the nonlinear transport term. For both cases, the reaction part has the form of the third order polynomial with three distinct roots. We found analytic one-soliton solutions and the relationships between three simple roots and the wave speed of the soliton. For the first case, if one of the roots is the mean value of other two roots, the soliton is static.We show that the restriction on three distinct roots to obtain moving soliton is removed in the second case by, adding nonlinear transport term to the first equation.Article Citation - WoS: 81Citation - Scopus: 79The Resonant Nonlinear Schrödinger Equation in Cold Plasma Physics. Application of Bäcklund-Darboux Transformations and Superposition Principles(Cambridge University Press, 2007) Lee, Jiunhung; Pashaev, Oktay; Rogers, Colin; Schief, W. K.A system of nonlinear equations governing the transmission of uni-axial waves in a cold collisionless plasma subject to a transverse magnetic field is reduced to the recently proposed resonant nonlinear Schrödinger (RNLS) equation. This integrable variant of the standard nonlinear Schrödinger equation admits novel nonlinear superposition principles associated with Bäcklund-Darboux transformations. These are used here, in particular, to construct analytic descriptions of the interaction of solitonic magnetoacoustic waves propagating through the plasma.Article Citation - WoS: 6Citation - Scopus: 6Relativistic Dnls and Kaup-Newell Hierarchy(Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine, 2017) Pashaev, Oktay; Lee, Jyh HaoBy the recursion operator of the Kaup-Newell hierarchy we construct the relativistic derivative NLS (RDNLS) equation and the corresponding Lax pair. In the nonrelativistic limit c → ∞ it reduces to DNLS equation and preserves integrability at any order of relativistic corrections. The compact explicit representation of the linear problem for this equation becomes possible due to notions of the q-calculus with two bases, one of which is the recursion operator, and another one is the spectral parameter. © 2017, Institute of Mathematics. All rights reserved.