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Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/3

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    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 Sciences
    String 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.
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    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 Economics
    We 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.
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    On maximal partial Latin hypercubes
    (Springer, 2023) Donovan, Diane M.; Grannell, Mike J.; Department of Mathematics; Yazıcı, Emine Şule; Department of Mathematics; College of Sciences
    A lower bound is presented for the minimal number of filled cells in a maximal partial Latin hypercube of dimension d and order n. The result generalises and extends previous results for d= 2 (Latin squares) and d= 3 (Latin cubes). Explicit constructions show that this bound is near-optimal for large n> d . For d> n , a connection with Hamming codes shows that this lower bound gives a related upper bound for the same quantity. The results can be interpreted in terms of independent dominating sets in certain graphs, and in terms of codes that have covering radius 1 and minimum distance at least 2.
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    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; 113760
    Optimization 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.
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    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/A
    In 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.
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    Symplectic and Lagrangian surfaces in 4-manifolds
    (Rocky Mt Math Consortium, 2008) Department of Mathematics; Etgü, Tolga; Faculty Member; Department of Mathematics; College of Sciences; 16206
    This is a brief summary of recent examples of isotopically different symplectic and Lagrangian surfaces representing a fixed homology class in a simply-connected symplectic 4-manifold.
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    Number of least area planes in gromov hyperbolic 3-spaces
    (American Mathematical Society (AMS), 2010) Department of Mathematics; Coşkunüzer, Barış; Faculty Member; Department of Mathematics; College of Sciences; N/A
    We show that for a generic simple closed curve Gamma in the asymptotic boundary of a Gromov hyperbolic 3-space with cocompact metric X, there exists a unique least area plane Sigma in X such that partial derivative(infinity)Sigma = Gamma. This result has interesting topological applications for constructions of canonical 2-dimensional objects in Gromov hyperbolic 3-manifolds.
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    Decomposition of lambda K-nu into kites and 4-cycles
    (Charles Babbage Research Centre, 2017) Milici, Salvatore; Department of Mathematics; Küçükçifçi, Selda; Faculty Member; Department of Mathematics; College of Sciences; 105252
    Given a collection of graphs H, an H-decomposition of λkv is a decomposition of the edges of λKv into isomorphic copies of graphs in Ti. A kite is a triangle with a tail consisting of a single edge. In this paper we investigate the decomposition problem when H is the set containing a kite and a 4-cycle, that is; this paper gives a complete solution to the problem of decomposing λKv into r kites and s 4-cycles for every admissible values of v, λ, r and s.
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    Locating a nearest matrix with an eigenvalue of prespecified algebraic multiplicity
    (Springer, 2011) N/A; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760
    The Wilkinson distance of a matrix A is the two-norm of the smallest perturbation E so that A + E has a multiple eigenvalue. Malyshev derived a singular value optimization characterization for the Wilkinson distance. In this work we generalize the definition of the Wilkinson distance as the two-norm of the smallest perturbation so that the perturbed matrix has an eigenvalue of prespecified algebraic multiplicity. We provide a singular value characterization for this generalized Wilkinson distance. Then we outline a numerical technique to solve the derived singular value optimization problems. In particular the numerical technique is applicable to Malyshev's formula to compute the Wilkinson distance as well as to retrieve a nearest matrix with a multiple eigenvalue.
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    Extension of one-dimensional proximity regions to higher dimensions
    (Elsevier, 2010) Department of Mathematics; Ceyhan, Elvan; Faculty Member; Department of Mathematics; College of Sciences; N/A
    Proximity regions (and maps) are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in various fields. The simplest of such maps is the spherical proximity map which gave rise to class cover catch digraph (CCCD) and was applied to pattern classification. In this article, we note some appealing properties of the spherical proximity map in compact intervals on the real line, thereby introduce the mechanism and guidelines for defining new proximity maps in higher dimensions. For non-spherical PCDs, Delaunay tessellation (triangulation in the real plane) is used to partition the region of interest in higher dimensions. We also introduce the auxiliary tools used for the construction of the new proximity maps, as well as some related concepts that will be used in the investigation and comparison of these maps and the resulting PCDs. We provide the distribution of graph invariants, namely, domination number and relative density, of the PCDs and characterize the geometry invariance of the distribution of these graph invariants for uniform data and provide some newly defined proximity maps in higher dimensions as illustrative examples. (C) 2010 Elsevier B.V. All rights reserved.