Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Conference Object Measure on Time Scales With Mathematica(Springer Verlag, 2006) Ufuktepe, Ünal; Yantır, AhmetIn this paper we study the Lebesgue Delta-measure on time scales. We refer to [3, 4] for the main notions and facts from the general measure and Lebesgue Delta integral theory. The objective of this paper is to show how the main concepts of Mathematica can be applied to fundamentals of Lebesgue Delta- and Lebesgue Delta- measure on an arbitrary time scale and also on a discrete time scale whose rule is given by the reader. As the time scale theory is investigated in two parts, by means of alpha and rho operators, we named the measures on time scales by the set function DMeasure and NMeasure respectively for arbitrary time scales.Article Citation - WoS: 8Citation - Scopus: 7Time-Evolution of Squeezed Coherent States of a Generalized Quantum Parametric Oscillator(American Institute of Physics, 2019) Atılgan Büyükaşık, Şirin; Çayiç, ZehraTime evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.Article Locally Isomorphic Torsionless Modules Over Domains of Finite Character(World Scientific Publishing, 2019) Saylam, Başak Ay; Klingler, LeeIn a 2002 paper, P. Goeters and B. Olberding compare local, near, and stable isomorphisms of torsionless modules over h-local domains. In this paper, we compare these weaker forms of isomorphisms of torsionless modules over domains of finite character.Article Citation - WoS: 8Citation - Scopus: 8On the Structure of Modules Defined by Subinjectivity(World Scientific Publishing, 2019) Altınay, Ferhat; Büyükaşık, Engin; Durgun, YılmazThe aim of this paper is to present new results and generalize some results about indigent modules. The commutative rings whose simple modules are indigent or injective are fully determined. The rings whose cyclic right modules are indigent are shown to be semisimple Artinian. We give a complete characterization of indigent modules over commutative hereditary Noetherian rings. We show that a reduced module is indigent if and only if it is a Whitehead test module for injectivity over commutative hereditary noetherian rings. Furthermore, Dedekind domains are characterized by test modules for injectivity by subinjectivity.Conference Object Dynamics of Squeezed States of a Generalized Quantum Parametric Oscillator(IOP Publishing, 2019) Atılgan Büyükaşık, Şirin; Çayiç, ZehraTime-evolution of squeezed coherent states of a generalized Caldirola-Kanai type quantum parametric oscillator is found explicitly using the exact evolution operator obtained by the Wei-Norman algebraic approach. Properties of these states are investigated according to the parameters of the unitary squeeze operator and the time-variable parameters of the generalized quadratic Hamiltonian. As an application, we consider exactly solvable quantum models with specific frequency modification for which the corresponding classical oscillator is in underdamping case and driving forces are of sinusoidal type. For each model we explicitly provide the evolution of the squeezed coherent states and discuss their behavior.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.Article Citation - WoS: 6Citation - Scopus: 7Integrable Systems From Inelastic Curve Flows in 2-And 3-Dimensional Minkowski Space(Taylor & Francis, 2016) Alkan, Kıvılcım; Anco, Stephen C.Integrable systems are derived from inelastic flows of timelike, spacelike, and null curves in 2-and 3- dimensional Minkowski space. The derivation uses a Lorentzian version of a geometrical moving frame method which is known to yield the modified Korteveg-de Vries (mKdV) equation and the nonlinear Schrodinger (NLS) equation in 2- and 3- dimensional Euclidean space, respectively. In 2-dimensional Minkowski space, time-like/space-like inelastic curve flows are shown to yield the defocusing mKdV equation and its bi-Hamiltonian integrability structure, while inelastic null curve flows are shown to give rise to Burgers' equation and its symmetry integrability structure. In 3-dimensional Minkowski space, the complex defocusing mKdV equation and the NLS equation along with their bi-Hamiltonian integrability structures are obtained from timelike inelastic curve flows, whereas spacelike inelastic curve flows yield an interesting variant of these two integrable equations in which complex numbers are replaced by hyperbolic (split-complex) numbers.Article On the Number of Bound States of Semirelativistic Hamiltonian With Dirac Delta Potentials in One Dimension(National Research Council of Canada, 2018) Erman, FatihWe study the bound state problem for semirelativistic N attractive Dirac delta-potentials in one dimension. We give a sufficient condition for the Hamiltonian to have N bound states and give an explicit criterion for it.