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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 Open Access Analysis of copositive optimization based linear programming bounds on standard quadratic optimization(Springer, 2015) Department of Industrial Engineering; Sağol, Gizem; Yıldırım, Emre Alper; Faculty Member; Department of Industrial Engineering; Graduate School of Sciences and Engineering; College of EngineeringThe problem of minimizing a quadratic form over the unit simplex, referred to as a standard quadratic optimization problem, admits an exact reformulation as a linear optimization problem over the convex cone of completely positive matrices. This computationally intractable cone can be approximated in various ways from the inside and from the outside by two sequences of nested tractable convex cones of increasing accuracy. In this paper, we focus on the inner polyhedral approximations due to YA +/- ldA +/- rA +/- m (Optim Methods Softw 27(1):155-173, 2012) and the outer polyhedral approximations due to de Klerk and Pasechnik (SIAM J Optim 12(4):875-892, 2002). We investigate the sequences of upper and lower bounds on the optimal value of a standard quadratic optimization problem arising from these two hierarchies of inner and outer polyhedral approximations. We give complete algebraic descriptions of the sets of instances on which upper and lower bounds are exact at any given finite level of the hierarchy. We identify the structural properties of the sets of instances on which upper and lower bounds converge to the optimal value only in the limit. We present several geometric and topological properties of these sets. Our results shed light on the strengths and limitations of these inner and outer polyhedral approximations in the context of standard quadratic optimization.Publication Open Access Averages of values of L-series(American Mathematical Society (AMS), 2013) Ono, Ken; Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803We obtain an exact formula for the average of values of L-series over two independent odd characters. The average of any positive moment of values at s = 1 is then expressed in terms of finite cotangent sums subject to congruence conditions. As consequences, bounds on such cotangent sums, limit points for the average of first moment of L-series at s = 1 and the average size of positive moments of character sums related to the class number are deduced.Publication Open Access Control of fork-join processing networks with multiple job types and parallel shared resources(The Institute for Operations Research and the Management Sciences (INFORMS), 2021) Department of Business Administration; Özkan, Erhun; Faculty Member; Department of Business Administration; College of Administrative Sciences and Economics; 294016A fork-join processing network is a queueing network in which tasks associated with a job can be processed simultaneously. Fork-join processing networks are prevalent in computer systems, healthcare, manufacturing, project management, justice systems, and so on. Unlike the conventional queueing networks, fork-join processing networks have synchronization constraints that arise because of the parallel processing of tasks and can cause significant job delays. We study scheduling in fork-join processing networks with multiple job types and parallel shared resources. Jobs arriving in the system fork into arbitrary number of tasks, then those tasks are processed in parallel, and then they join and leave the network. There are shared resources processing multiple job types. We study the scheduling problem for those shared resources (i.e., which type of job to prioritize at any given time) and propose an asymptotically optimal scheduling policy in diffusion scale.Publication Metadata only Matrix polynomials with specified eigenvalues(Elsevier Science Inc, 2015) Karow, Michael; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760This work concerns the distance in the 2-norm from a given matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane. Initially, we consider perturbations of the constant coefficient matrix only. A singular value optimization characterization is derived for the associated distance. We also consider the distance in the general setting, when all of the coefficient matrices are perturbed. In this general setting, we obtain a lower bound in terms of another singular value optimization problem. The singular value optimization problems derived facilitate the numerical computation of the distances.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 Open Access Nearest linear systems with highly deficient reachable subspaces(Society for Industrial and Applied Mathematics (SIAM), 2012) Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760We consider the 2-norm distance tau(r)(A, B) from a linear time-invariant dynamical system (A, B) of order n to the nearest system (A + Delta A(*), B + Delta B-*) whose reachable subspace is of dimension r < n. We first present a characterization to test whether the reachable subspace of the system has dimension r, which resembles and can be considered as a generalization of the Popov-Belevitch-Hautus test for controllability. Then, by exploiting this generalized Popov-Belevitch-Hautus characterization, we derive the main result of this paper, which is a singular value optimization characterization for tau(r)(A, B). A numerical technique to solve the derived singular value optimization problems is described. The numerical results on a few examples illustrate the significance of the derived singular value characterization for computational purposes.Publication Open Access Nonlinear eigenvalue problems with specified eigenvalues(Society for Industrial and Applied Mathematics (SIAM), 2014) Karow, Michael; Kressner, Daniel; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760This work considers eigenvalue problems that are nonlinear in the eigenvalue parameter. Given such a nonlinear eigenvalue problem T, we are concerned with finding the minimal backward error such that T has a set of prescribed eigenvalues with prescribed algebraic multiplicities. We consider backward errors that only allow constant perturbations, which do not depend on the eigenvalue parameter. While the usual resolvent norm addresses this question for a single eigenvalue of multiplicity one, the general setting involving several eigenvalues is significantly more difficult. Under mild assumptions, we derive a singular value optimization characterization for the minimal perturbation that addresses the general case.Publication Open Access Numerical optimization of eigenvalues of Hermitian matrix functions(Society for Industrial and Applied Mathematics (SIAM), 2014) Department of Mathematics; N/A; Mengi, Emre; Yıldırım, Emre Alper; Kılıç, Mustafa; Faculty Member; Faculty Member; Department of Mathematics; College of Sciences; College of Engineering; 113760; N/A; N/AThis work concerns the global minimization of a prescribed eigenvalue or a weighted sum of prescribed eigenvalues of a Hermitian matrix-valued function depending on its parameters analytically in a box. We describe how the analytical properties of eigenvalue functions can be put into use to derive piecewise quadratic functions that underestimate the eigenvalue functions. These piecewise quadratic underestimators lead us to a global minimization algorithm, originally due to Breiman and Cutler. We prove the global convergence of the algorithm and show that it can be effectively used for the minimization of extreme eigenvalues, e.g., the largest eigenvalue or the sum of the largest specified number of eigenvalues. This is particularly facilitated by the analytical formulas for the first derivatives of eigenvalues, as well as analytical lower bounds on the second derivatives that can be deduced for extreme eigenvalue functions. The applications that we have in mind also include the H-infinity-norm of a linear dynamical system, numerical radius, distance to uncontrollability, and various other nonconvex eigenvalue optimization problems, for which, generically, the eigenvalue function involved is simple at all points.Publication Metadata only On the well-posedness of the hyperelastic rod equation(Springer Heidelberg, 2019) N/A; Department of Mathematics; İnci, Hasan; Faculty Member; Department of Mathematics; College of Sciences; 274184In this paper we consider the hyperelastic rod equation on the Sobolev spaces Hs(R), s>3/2. Using a geometric approach we show that for any T>0 the corresponding solution map, u(0)?u(T), is nowhere locally uniformly continuous. The method applies also to the periodic case Hs(T), s>3/2.