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.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.Research Project Genel Dispersiyona Sahip Doğrusal Olmayan İntegrallenebilen Sistemler ve Dinamik Kuantum Simetrileri(2019) Pashaev, Oktay; Büyükaşık, Şirin AtılganBu projede, keyfi ve q-deforme olmuş dispersiyona sahip doğrusal olmayan yeni integrallenebilir sistemler ve tam çözülebilen dinamik quantum simetrilerine sahip kuantum modellerin hiyerarşisi inşa edildi. İlk olarak, normal koordinatlarda q- ve f- deforme olmuş osilatörler de dahil olmak üzere, klasik çok boyutlu integrallenebilen sistemler deforme olmuş keyfi deformasyona sahip doğrusal olmayan osilatörler olarak çözüldü. Schrödinger gösteriminde, keyfi dispersiyona sahip quantum parametrik osilatör denklemi çözüldü, quantum dinamik simetrileri bulundu, zamana bağlı evrim ve tam çözümleri de incelendi. Doğrusal olmayan integrallenebilen evrim denklemleri NLS, DNLS ve AKNS ile onların doğrusal gösterimleri için, doğrusal olmayan deformasyona sahip dispersiyonlar inşa edildi. Özel olarak, yineleme operatörün yardımıyla, q-deforme olmuş ve göreli dispersiyona sahip NLS, DNLS denklemler hiyerarşisi ve karşılık gelen rezonant soliton denklemleri elde edildi. Dinamik simetri ve evrim operatörü yöntemleri ile zamana bağlı ağırlık ve frekansa sahip kuantum parametrik osilatör için Schrödinger denklemi çözüldü. Bu modeller için koherent durumlar, sıkıştırılmış koherent durumlar, resonant ve sönümleme dinamikleri elde edildi. Kuantum Fourier dönüşümü yardımıyla, birin kökleri olan q ile ilişkili, kuantum grup simetrisi olarak koherent durumların superpozisyonu inşa edildi. Bu kaleydoskop durumlar kuantum enformasyon birimi olarak görülebilir. Tekli ve çoklu kubitler için Apollonius gösterimi bulundu. Halka biçimli bölgede N-poligon girdaplar ve onların doğrusal olmayan osilatör olarak quantizasyonu çalışıldı. Generik pq-Fibonacci ve altın analitik durumlar için q-analitik koherent durumlar ve ilgili Fock-Bargman gösterimleri tanıtıldı.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 - WoS: 2Citation - Scopus: 2Quantum Group Symmetry for Kaleidoscope of Hydrodynamic Images and Quantum States(IOP Publishing, 2019) Pashaev, OktayThe hydrodynamic flow in several bounded domains can be formulated by the image theorems, like the two circle, the wedge and the strip theorems, describing flow by q-periodic functions. Depending on geometry of the domain, parameter q has different geometrical meanings and values. In the special case of the wedge domain, with q as a primitive root of unity, the set of images appears as a regular polygon kaleidoscope. By interpreting the wave function in the Fock-Barman representation as complex potential of a flow, we find modn projection operators in the space of quantum coherent states, related with operator q-numbers. They determine the units of quantum information as kaleidoscope of quantum states with quantum group symmetry of the q-oscillator. Expansion of Glauber coherent states to these units and corresponding entropy are discussed.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 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 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.Conference Object Citation - WoS: 5Citation - Scopus: 6Quantum Calculus of Classical Vortex Images, Integrable Models and Quantum States(IOP Publishing Ltd., 2016) Pashaev, OktayFrom two circle theorem described in terms of q-periodic functions, in the limit q→1 we have derived the strip theorem and the stream function for N vortex problem. For regular N-vortex polygon we find compact expression for the velocity of uniform rotation and show that it represents a nonlinear oscillator. We describe q-dispersive extensions of the linear and nonlinear Schrodinger equations, as well as the q-semiclassical expansions in terms of Bernoulli and Euler polynomials. Different kind of q-analytic functions are introduced, including the pq-analytic and the golden analytic functions.