Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
Browse
55 results
Search Results
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.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: 3Citation - Scopus: 4Quantum Coin Flipping, Qubit Measurement, and Generalized Fibonacci Numbers(Pleiades Publishing, 2021) Pashaev, OktayThe problem of Hadamard quantum coin measurement in n trials, with an arbitrary number of repeated consecutive last states, is formulated in terms of Fibonacci sequences for duplicated states, Tribonacci numbers for triplicated states, and N-Bonacci numbers for arbitrary N-plicated states. The probability formulas for arbitrary positions of repeated states are derived in terms of the Lucas and Fibonacci numbers. For a generic qubit coin, the formulas are expressed by the Fibonacci and more general, N-Bonacci polynomials in qubit probabilities. The generating function for probabilities, the Golden Ratio limit of these probabilities, and the Shannon entropy for corresponding states are determined. Using a generalized Born rule and the universality of the n-qubit measurement gate, we formulate the problem in terms of generic n- qubit states and construct projection operators in a Hilbert space, constrained on the Fibonacci tree of the states. The results are generalized to qutrit and qudit coins described by generalized FibonacciN-Bonacci sequences.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: 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.