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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 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 Metadata only Big Heegner point Kolyvagin system for a family of modular forms(Springer International Publishing Ag, 2014) Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of Sciences; N/AThe principal goal of this paper is to develop Kolyvagin's descent to apply with the big Heegner point Euler system constructed by Howard for the big Galois representation attached to a Hida family of elliptic modular forms. In order to achieve this, we interpolate and control the Tamagawa factors attached to each member of the family at bad primes, which should be of independent interest. Using this, we then work out the Kolyvagin descent on the big Heegner point Euler system so as to obtain a big Kolyvagin system that interpolates the collection of Kolyvagin systems obtained by Fouquet for each member of the family individually. This construction has standard applications to Iwasawa theory, which we record at the end.Publication Open Access Boxlib with tiling: an adaptive mesh refinement software framework(Society for Industrial and Applied Mathematics (SIAM), 2016) Zhang, W.; Almgren, A.; Day, M.; Nguyen, T.; Shalf, J.; Department of Computer Engineering; Erten, Didem Unat; Faculty Member; Department of Computer Engineering; College of Engineering; 219274In this paper we introduce a block-structured adaptive mesh refinement software framework that incorporates tiling, a well-known loop transformation. Because the multiscale, multiphysics codes built in boxlib are designed to solve complex systems at high resolution, performance on current and next generation architectures is essential. With the expectation of many more cores per node on next generation architectures, the ability to effectively utilize threads within a node is essential, and the current model for parallelization will not be sufficient. We describe a new version of boxlib in which the tiling constructs are embedded so that boxlib-based applications can easily realize expected performance gains without extra effort on the part of the application developer. We also discuss a path forward to enable future versions of boxlib to take advantage of NUMA-aware optimizations using the tida portable library.Publication Open Access Computation of the canonical lifting via division polynomials(Unión Matemática Argentina, 2015) Department of Mathematics; Erdoğan, Altan; Teaching Faculty; Department of Mathematics; College of SciencesWe study the canonical lifting of ordinary elliptic curves over the ring of Witt vectors. We prove that the canonical lifting is compatible with the base field of the given ordinary elliptic curve which was first proved in Finotti, J. Number Theory 130 (2010), 620-638. We also give some results about division polynomials of elliptic curves de fined over the ring of Witt vectors.Publication Metadata only Computation of the canonical lifting via division polynomials(Union Matematica Argentina, 2015) Department of Mathematics; Erdoğan, Altan; Teaching Faculty; Department of Mathematics; College of Sciences; N/AWe study the canonical lifting of ordinary elliptic curves over the ring of Witt vectors. We prove that the canonical lifting is compatible with the base field of the given ordinary elliptic curve which was first proved in Finotti, J. Number Theory 130 (2010), 620-638. We also give some results about division polynomials of elliptic curves de fined over the ring of Witt vectors.Publication Open Access Defining sets of full designs with block size three II(Springer, 2012) Donovan, Diane; Lefevre, James; Waterhouse, Mary; Department of Mathematics; Yazıcı, Emine Şule; Faculty Member; Department of Mathematics; College of Sciences; 27432A defining set of a t-(v, k, lambda) design is a subcollection of its blocks which is contained in a unique t-design with the given parameters. A minimal defining set is a defining set, none of whose proper subcollections is a defining set. The spectrum of minimal defining sets of a design D is the set {|M| | M is a minimal defining set of D}. The unique simple design with parameters is said to be the full design on v elements. This paper studies the minimal defining sets of full designs when t = 2 and k = 3. The largest known minimal defining set is given. The existence of a continuous section of the spectrum comprising asymptotically 9v (2)/50 values is shown. This gives a quadratic length section of continuous spectrum where only a linear section with respect to v was known before.Publication Metadata only 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/AProximity 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.Publication Metadata only 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; 113760The 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.Publication Open Access Maximum uniformly resolvable decompositions of K-v and K-v - I into 3-stars and 3-cycles(Elsevier, 2015) Milici, Salvatore; Tuza, Zsolt; Department of Mathematics; Küçükçifçi, Selda; Faculty Member; Department of Mathematics; College of Sciences; 105252Let K-v denote the complete graph of order v and K-v - I denote K-v minus a 1-factor. In this article we investigate uniformly resolvable decompositions of K-v and K-v - I into r classes containing only copies of 3-stars and s classes containing only copies of 3-cycles. We completely determine the spectrum in the case where the number of resolution classes of 3-stars is maximum.