Researcher: Doğan, Keremcan
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Doğan, Keremcan
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Publication Metadata only ‘Anti-commutable’ local pre-Leibniz algebroids and admissible connections(Elsevier, 2023) Department of Physics; N/A; Dereli, Tekin; Doğan, Keremcan; Faculty Member; PhD Student; Department of Physics; College of Sciences; Graduate School of Sciences and Engineering; 201358; N/AThe concept of algebroid is convenient as a basis for constructions of geometrical frameworks. For example, metric-affine and generalized geometries can be written on Lie and Courant algebroids, respectively. Furthermore, string theories might make use of many other algebroids such as metric algebroids, higher Courant algebroids, or conformal Courant algebroids. Working on the possibly most general algebroid structure, which generalizes many of the algebroids used in the literature, is fruitful as it creates a chance to study all of them at once. Local pre-Leibniz algebroids are such general ones in which metric-connection geometries are possible to construct. On the other hand, the existence of the 'locality operator', which is present for the left-Leibniz rule for the bracket, necessitates the modification of torsion and curvature operators in order to achieve tensorial quantities. In this paper, this modification of torsion and curvature is explained from the point of view that the modification is applied to the bracket instead. This leads one to consider 'anti-commutable' local pre-Leibniz algebroids which satisfy an anti-commutativity-like property defined with respect to a choice of an equivalence class of connections. These 'admissible' connections are claimed to be the necessary ones while working on a geometry of algebroids. This claim is due to the fact that one can prove many desirable properties and relations if one uses only admissible connections. For instance, for admissible connections, we prove the first and second Bianchi identities, Cartan structure equations, Cartan magic formula, the construction of Levi-Civita connections, the decomposition of connection in terms of torsion and non-metricity. These all are possible because the modified bracket becomes anti-symmetric for an admissible connection so that one can apply the machinery of almost-or pre-Lie algebroids. We investigate various algebroid structures from the literature and show that they admit admissible connections which are metric-compatible in some generalized sense. Moreover, we prove that local pre-Leibniz algebroids that are not anti-commutable cannot be equipped with a torsion-free, and in particular Levi-Civita, connection.Publication Metadata only Metric-connection geometries on pre-Leibniz algebroids: a search for geometrical structure in string models(Amer Inst Physics, 2021) Department of Physics; N/A; Dereli, Tekin; Doğan, Keremcan; Faculty Member; PhD Student; Department of Physics; College of Sciences; Graduate School of Sciences and Engineering; 201358; N/AMetric-affine and generalized geometries are arguably the appropriate mathematical frameworks for Einstein's theory of gravity and low-energy effective string field theory, respectively. In fact, mathematical structures in a metric-affine geometry are constructed on the tangent bundle, which is itself a Lie algebroid, whereas those in generalized geometries, which form the basis of double field theories, are constructed on Courant algebroids. Lie, Courant, and higher Courant algebroids, which are used in exceptional field theories, are all known to be special cases of pre-Leibniz algebroids. As mathematical structures on these algebroids are essential in string models, it is natural to work on a more unifying geometrical framework. Provided with some additional ingredients, the construction of such geometries can all be carried over to regular pre-Leibniz algebroids. We define below the notions of locality structures and locality projectors, which are some necessary ingredients. In terms of these structures, E-metric-connection geometries are constructed with (possibly) a minimum number of assumptions. Certain small gaps in the literature are also filled as we go along. E-Koszul connections, as a generalization of Levi-Civita connections, are defined and shown to be helpful for some results including a simple generalization of the fundamental theorem of Riemannian geometry. The existence and non-existence of E-Levi-Civita connections are discussed for certain cases. We also show that metric-affine geometries can be constructed in a unique way as special cases of E-metric-connection geometries. Some aspects of Lie and Lie-type algebroids are studied, where the latter are defined here as a generalization of Lie algebroids. Moreover, generalized geometries are shown to follow as special cases, and various properties of linear generalized-connections are proven in the present framework. Similarly, uniqueness of the locality projector in the case of exact Courant algebroids is proven, a result that explains why the curvature operator, defined with a projector in the double field theory literature, is a necessity.Publication Metadata only Statistical geometry and Hessian structures on pre-Leibniz algebroids(IOP Publishing Ltd, 2022) Department of Physics; Doğan, Keremcan; PhD Student; Department of Physics; Graduate School of Sciences and Engineering; N/AWe introduce statistical, conjugate connection and Hessian structures on anti-commutable pre-Leibniz algebroids. Anti-commutable pre-Leibniz algebroids are special cases of local pre-Leibniz algebroids, which are still general enough to include many physically motivated algebroids such as Lie, Courant, metric and higher-Courant algebroids. They create a natural framework for generalizations of differential geometric structures on a smooth manifold. The symmetrization of the bracket on an anti-commutable pre-Leibniz algebroid satisfies a certain property depending on a choice of an equivalence class of connections which are called admissible. These admissible connections are shown to be necessary to generalize aforementioned structures on pre-Leibniz algebroids. Consequently, we prove that, provided certain conditions are met, statistical and conjugate connection structures are equivalent when defined for admissible connections. Moreover, we also show that for 'projected-torsion-free' connections, one can generalize Hessian metrics and Hessian structures. We prove that any Hessian structure yields a statistical structure, where these results are completely parallel to the ones in the manifold setting. We also prove a mild generalization of the fundamental theorem of statistical geometry. Moreover, we generalize a-connections, strongly conjugate connections and relative torsion operator, and prove some analogous results. © 2021 Published under licence by IOP Publishing Ltd.Publication Metadata only Spectral singularities, threshold gain, and output intensity for a slab laser with mirrors(Academic Press Inc Elsevier Science, 2018) Department of Physics; Department of Mathematics; Department of Mathematics; Doğan, Keremcan; Mostafazadeh, Ali; Sarısaman, Mustafa; PhD Student; Faculty Member; Researcher; Department of Physics; Department of Mathematics; Graduate School of Sciences and Engineering; College of Sciences; College of Sciences; N/A; 4231; 217652We explore the consequences of the emergence of linear and non-linear spectral singularities in TE modes of a homogeneous slab of active optical material that is placed between two mirrors. We use the results together with two basic postulates regarding the behavior of laser light emission to derive explicit expressions for the laser threshold condition and output intensity for these modes of the slab and discuss their physical implications. In particular, we reveal the details of the dependence of the threshold gain and output intensity on the position and properties of the mirrors and on the real part of the refractive index of the gain material. (C) 2018 Elsevier Inc. All rights reserved.