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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 On the past, present, and future of the Diebold-Yilmaz approach to dynamic network connectedness(Elsevier Science Sa, 2023) Diebold, Francis X.; Department of Economics; Yılmaz, Kamil; Department of Economics; College of Administrative Sciences and EconomicsWe offer retrospective and prospective assessments of the Diebold-Yilmaz connected-ness research program, combined with personal recollections of its development. Its centerpiece in many respects is Diebold and Yilmaz (2014), around which our discussion is organized.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 On the network topology of variance decompositions: measuring the connectedness of financial firms (Reprinted from Journal of Econometrics, Vol 182, Issue 1, September 2014, Pages 119-134)(Elsevier Science Sa, 2023) Diebold, Francis X.; Department of Economics; Yılmaz, Kamil; Department of Economics; College of Administrative Sciences and EconomicsWe propose several connectedness measures built from pieces of variance decomposi-tions, and we argue that they provide natural and insightful measures of connectedness. We also show that variance decompositions define weighted, directed networks, so that our connectedness measures are intimately related to key measures of connectedness used in the network literature. Building on these insights, we track daily time-varying connectedness of major U.S. financial institutions' stock return volatilities in recent years, with emphasis on the financial crisis of 2007-2008.Publication Metadata only A characterization of heaviness in terms of relative symplectic cohomology(Wiley, 2024) Mak, Cheuk Yu; Sun, Yuhan; Department of Mathematics; Varolgüneş, Umut; Department of Mathematics; College of SciencesFor a compact subset K$K$ of a closed symplectic manifold (M,omega)$(M, \omega)$, we prove that K$K$ is heavy if and only if its relative symplectic cohomology over the Novikov field is nonzero. As an application, we show that if two compact sets are not heavy and Poisson commuting, then their union is also not heavy. A discussion on superheaviness together with some partial results is also included.Publication Metadata only Stopping levels for a spectrally negative Markov additive process(Springer Science and Business Media Deutschland GmbH, 2024) Vardar-Acar, C.; Department of Mathematics; Çağlar, Mine; Department of Mathematics; College of SciencesThe optimal stopping problem for pricing Russian options in finance requires taking the supremum of the discounted reward function over all finite stopping times. We assume the logarithm of the asset price is a spectrally negative Markov additive process with finitely many regimes. The reward function is given by the exponential of the running supremum of the price process. Previous work on Russian optimal stopping problem suggests that the optimal stopping time would be an upcrossing time of the drawdown at a certain level for each regime. We derive explicit formulas for identifying the stopping levels and computing the corresponding value functions through a recursive algorithm. A numerical is provided for finding these stopping levels and their value functions.Publication Metadata only Counterexamples to regularity of Mane projections in the theory of attractors(Institute of Physics (IOP) Publishing, 2013) Eden, Alp; Zelik, Sergey V.; Department of Mathematics; Kalantarov, Varga; Faculty Member; Department of Mathematics; College of Sciences; 117655This paper is a study of global attractors of abstract semilinear parabolic equations and their embeddings in finite-dimensional manifolds. As is well known, a sufficient condition for the existence of smooth (at least C-1-smooth) finite-dimensional inertial manifolds containing a global attractor is the so-called spectral gap condition for the corresponding linear operator. There are also a number of examples showing that if there is no gap in the spectrum, then a C-1-smooth inertial manifold may not exist. on the other hand, since an attractor usually has finite fractal dimension, by Mane's theorem it projects bijectively and Holder-homeomorphically into a finite-dimensional generic plane if its dimension is large enough. It is shown here that if there are no gaps in the spectrum, then there exist attractors that cannot be embedded in any Lipschitz or even log-Lipschitz finite-dimensional manifold. Thus, if there are no gaps in the spectrum, then in the general case the inverse Mane projection of the attractor cannot be expected to be Lipschitz or log-Lipschitz. Furthermore, examples of attractors with finite Hausdorff and infinite fractal dimension are constructed in the class of non-linearities of finite smoothness.Publication Metadata only A support function based algorithm for optimization with eigenvalue constraints(Siam Publications, 2017) N/A; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalue functions and is of practical interest because of a wide range of applications in fields such as structural design and control theory. Here we focus on the optimization of a linear objective subject to a constraint on the smallest eigenvalue of an analytic and Hermitian matrix-valued function. We propose a numerical approach based on quadratic support functions that overestimate the smallest eigenvalue function globally. the quadratic support functions are derived by employing variational properties of the smallest eigenvalue function over a set of Hermitian matrices. We establish the local convergence of the algorithm under mild assumptions and deduce a precise rate of convergence result by viewing the algorithm as a fixed point iteration. the convergence analysis reveals that the algorithm is immune to the nonsmooth nature of the smallest eigenvalue. We illustrate the practical applicability of the algorithm on the pseudospectral functions.Publication Metadata only Continuous dependence for the convective brinkman–forchheimer equations(Taylor & Francis, 2005) Çelebi, A.O.; Ugurlu, D.; Department of Mathematics; Kalantarov, Varga; Faculty Member; Department of Mathematics; College of Sciences; 117655In this article, we have considered the convective Brinkman–Forchheimer equations with Dirichlet's boundary conditions. The continuous dependence of solutions on the Forchheimer coefficient in H 1 norm is proved.Publication Metadata only On the anticyclotomic Iwasawa theory of CM forms at supersingular primes(European Mathematical Soc, 2015) Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of Sciences; N/AIn this paper, we study the anticyclotomic Iwasawa theory of a CM form f of even weight w >= 2 at a supersingular prime, generalizing the results in weight 2, due to Agboola and Howard. In due course, we are naturally lead to a conjecture on universal norms that generalizes a theorem of Perrin-Riou and Berger and another that generalizes a conjecture of Rubin (the latter seems linked to the local divisibility of Heegner points). Assuming the truth of these conjectures, we establish a formula for the variation of the sizes of the Selmer groups attached to the central critical twist of f as one climbs up the anticyclotomic tower. We also prove a statement which may be regarded as a form of the anticyclotomic main conjecture (without p-adic L-functions) for the central critical twist of f.