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Publication Metadata only Metric-bourbaki algebroids: cartan calculus for m-theory(Elsevier, 2024) Çatal-Özer, Aybike; Doğan, Keremcan; Department of Physics; Dereli, Tekin; Department of Physics; College of SciencesString and M theories seem to require generalizations of usual notions of differential geometry on smooth manifolds. Such generalizations usually involve extending the tangent bundle to larger vector bundles equipped with various algebroid structures such as Courant algebroids, higher Courant algebroids, metric algebroids, or G-algebroids. The most general geometric scheme is not well understood yet, and a unifying framework for such algebroid structures is needed. Our aim in this paper is to propose such a general framework. Our strategy is to follow the hierarchy of defining axioms for a Courant algebroid: almostCourant - metric - pre -Courant - Courant. In particular, we focus on the symmetric part of the bracket and the metric invariance property, and try to make sense of them in a manner as general as possible. These ideas lead us to define new algebroid structures which we dub Bourbaki and metric-Bourbaki algebroids, together with their almostand pre -versions. For a special case of metric-Bourbaki algebroids that we call exact, we construct a collection of maps which generalize the Cartan calculus of exterior derivative, Lie derivative and interior product. This is done by a kind of reverse -mathematical analysis of the Severa classification of exact Courant algebroids. By abstracting crucial properties of this collection of maps, we define the notion of Bourbaki calculus. Conversely, given an arbitrary Bourbaki calculus, we construct a metric-Bourbaki algebroid by building up a standard bracket that is analogous to the Dorfman bracket. Moreover, we prove that any exact metric-Bourbaki algebroid satisfying some further conditions has to have a bracket that is the twisted version of the standard bracket; a partly analogous result to Severa classification. We prove that many physically and mathematically motivated algebroids from the literature are examples of these new algebroids, and when possible we construct a Bourbaki calculus on them. In particular, we show that the Cartan calculus can be seen as the Bourbaki calculus corresponding to an exact higher Courant algebroid. We also point out examples of Bourbaki calculi including the generalization of the Cartan calculus on vector bundle valued forms. One straightforward generalization of our constructions might be done by replacing the tangent bundle with an arbitrary Lie algebroid A. This step allows us to define an extension of our results, A -version, and extend our main results for them while proving many other algebroids from the literature fit into this framework.Publication Metadata only Temporal evolution of entropy and chaos in low amplitude seismic wave prior to an earthquake(Pergamon-Elsevier Science Ltd, 2023) Akilli, Mahmut; Ak, Mine; Department of Physics; Yılmaz, Nazmi; Department of Physics; College of SciencesThis study investigates the temporal changes of entropy and chaos in low-amplitude continuous seismic wave data prior to two moderate-level earthquakes. Specifically, we examine seismic signals before and during the Istanbul-Turkey earthquake of September 26, 2019 (M = 5.7), and the Duzce-Turkey earthquake of November 17, 2021 (M = 5.2), which occurred near the Marmara Sea region on the north-Anatolian fault line. We aim to identify changes in complexity and chaotic characteristics in the pre-earthquake seismic waves and explore the possibility of earthquake forecasting minutes before an earthquake. To accomplish this, we utilize windowed scalogram entropy and sample entropy methods and compared the results with Lyapunov exponents and windowed scale index. Our findings indicate that measuring the temporal change of entropy using windowed scalogram entropy is sensitive to the change in complexity due to the frequency shifts during the weak ground motion approaching an earthquake.On the other hand, Lyapunov exponents and sample entropy appear more effective in their response to the change in complexity and chaotic characteristics due to the change in the signal amplitude. Additionally, the windowed scale index can detect temporal fluctuations in the aperiodicity of the signal. Overall, our results suggest that all four methods can be valuable in characterizing complexity and chaos in short-time pre -earthquake seismic signals, differentiating earthquakes, and contributing to the development of earthquake forecasting techniques.Publication Metadata only A remarkable dynamical symmetry of the Landau problem(IOP Publishing Ltd, 2022) Nounahon, Philippe; Popov, Todor; Department of Physics; Dereli, Tekin; Faculty Member; Department of Physics; College of Sciences; 201358We show that the dynamical group of an electron in a constant magnetic feld is the group of symplectomorphisms Sp(4, R). It is generated by the spinorial realization of the conformal algebra so(2,3) considered in Dirac's seminal paper "A Remarkable Representation of the 3 + 2 de Sitter Group". The symplectic group Sp(4,R) is the double covering of the conformal group SO(2,3) of 2+1 dimensional Minkowski spacetime which is in turn the dynamical group of a hydrogen atom in 2 space dimensions. The Newton-Hooke duality between the 2D hydrogen atom and the Landau problem is explained via the Tits-Kantor-Koecher construction of the conformal symmetries of the Jordan algebra of real symmetric 2 × 2 matrices. The connection between the Landau problem and the 3D hydrogen atom is elucidated by the reduction of a Dirac spinor to a Majorana one in the Kustaanheimo-Stiefel spinorial regularization. © 2021 Published under licence by IOP Publishing Ltd.Publication Metadata only A trip around octonions(IOP Publishing Ltd, 2022) Bilge, Ayse Humeyra; Kocak, Sahin; Department of Physics; Dereli, Tekin; Faculty Member; Department of Physics; College of Sciences; 201358In these expository notes, after a contemplation on the dawn of octonions, we give proofs for the Frobenius theorem and the Hurwitz theorem, we review the basics of Clifford algebras and spin groups, and exemplify the startling role played by the octonions in 7- A nd 8-dimensional phenomena such as the special 3- A nd 4-forms, the Bonan form, Spin(7) and Spin(8) groups and the mysterious triality. © 2021 Published under licence by IOP Publishing Ltd.Publication Metadata only Maximal linear subspaces of strong self-dual 2-forms and the Bonan 4-form(Elsevier, 2011) Bilge, Ayşe Humeyra; Koçak, Şahin; Department of Physics; Dereli, Tekin; Faculty Member; Department of Physics; College of Sciences; 201358The notion of self-duality of 2-forms in 4-dimensions plays an eminent role in many areas of mathematics and physics, but although the 2-forms have a genuine meaning related to curvature and gauge-field-strength in higher dimensions also, their "self-duality" is something which is almost avoided above 4-dimensions. We show that self-duality of 2-forms is a very natural notion in higher (even) dimensions also and we prove the equivalence of some scattered and rarely used definitions in the literature. We demonstrate the usefulness of this higher self-duality by studying it in 8-dimensions and we derive a natural expression for the Bonan form in terms of self-dual 2-forms and we give an explicit expression of the local action of SO(8) on the Bonan form. (C) 2010 Elsevier Inc. All rights reserved.Publication Metadata only The 'wavelet' entropic index q of non-extensive statistical mechanics and superstatistics(2021) Akıllı, Mahmut; Akdeniz, K. Gediz; Department of Physics; Yılmaz, Nazmi; Teaching Faculty; Department of Physics; College of Sciences; 178427Generalized entropies developed for non-extensive statistical mechanics are derived from the Boltzmann-Gibbs-Shannon entropy by a real number q that is a parameter based on q-calculus; where is called ‘the entropic index’ and determines the degree of non-extensivity of a system in the interval between 1 and 3. In a very recent study, we introduced a new calculation method of the entropic index of non-extensive statistical mechanics. In this study, we show the mathematical proof of this calculation method of the entropic index. Firstly, we propose that the number of degrees of freedom, is proportional to the inverse of the wavelet scale index, where is a wavelet based parameter called wavelet scale index that quantitatively measures the non-periodicity of a signal in the interval between 0 and 1. Then, by applying this proposition to the superstatistics approach, we derive the equation that expresses the relationship between the entropic index and the wavelet scale index. Therefore, we name this -index as the ‘wavelet’ entropic index. Lastly, we calculate the Abe entropy, Landsberg-Vedral entropy and q-qualities of the Tsallis entropy of the Logistic Map and Hennon Map using the ‘wavelet’ entropic index, and based on our results, compare and discuss these generalized entropies.Publication Metadata only Laser beam propagation in a thermally loaded absorber(Optica Publishing Group, 1996) Department of Physics; Department of Mathematics; Department of Mathematics; Sennaroğlu, Alphan; Aşkar, Attila; Atay, Fatihcan; Faculty Member; Faculty Member; Faculty Member; Department of Physics; Department of Mathematics; College of Sciences; College of Sciences; College of Sciences; 23851; 178822; 253074Beam propagation in a thermally loaded absorber is analyzed by a novel method. The formulation identifies a dimensionless parameter controlling the strength of thermal effects.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 Degenerate spin groups as semi-direct products(Springer, 2010) Kocak, Sahin; Limoncu, Murat; Department of Physics; Dereli, Tekin; Faculty Member; Department of Physics; College of Sciences; 201358Let Q be a symmetric bilinear form on R(n)=R(p+q+r) with corank r, rank p+q and signature type (p, q), p resp. q denoting positive resp. negative dimensions. We consider the degenerate spin group Spin(Q) = Spin(p, q, r) in the sense of Crumeyrolle and prove that this group is isomorphic to the semi-direct product of the nondegenerate and indefinite spin group Spin(p, q) with the additive matrix group Mat (p + q, r)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.