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Publication Metadata only Deformations of Bloch groups and Aomoto dilogarithms in characteristic p(Academic Press Inc Elsevier Science, 2011) Department of Mathematics; Ünver, Sinan; Faculty Member; Department of Mathematics; College of Sciences; 177871In this paper, we study the Bloch group B-2(F[epsilon](2)) over the ring of dual numbers of the algebraic closure of the field with p elements, for a prime p >= 5. We show that a slight modification of Kontsevich's 11/2-logarithm defines a function on B-2(F[epsilon](2)). Using this function and the characteristic p version of the additive dilogarithm function that we previously defined, we determine the structure of the infinitesimal part of B-2(Ff[epsilon](2)) completely. This enables us to define invariants on the group of deformations of Aomoto dilogarithms and determine its structure. This final result might be viewed as the analog of Hilbert's third problem in characteristic p.Publication Metadata only The triangle intersection problem for K4 - E designs(Utilitas Mathematica Publishing Inc., 2007) Billington, Elizabeth J.; Lindner, C. C.; Department of Mathematics; Yazıcı, Emine Şule; Faculty Member; Department of Mathematics; College of Sciences; 27432An edge-disjoint decomposition of the complete graph Kn into copies of K4 - e, the simple graph with four vertices and five edges, is known to exist if and only if n ≡ 0 or 1 (mod 5) and n ≥ 6 (Bermond and Schönheim, Discrete Math. 19 (1997)). The intersection problem for K4 - e designs has also been solved (Billington, M. Gionfriddo and Lindner, J. Statist. Planning Inference 58 (1997)); this problem finds the number of common K4 - e blocks which two K4 - e designs on the same set may have. Here we answer the question: how many common triangles may two K4 - e designs on the same set have? Since it is possible for two K4 - e designs on the same set to have no common K4 - e blocks and yet some positive number of common triangles, this problem is largely independent of the earlier K4 - e intersection result.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 Addendum to “On the mean square average of special values of L-functions” [J. Number Theory 131 (8) (2011) 1470–1485](Elsevier, 2011) Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803Publication Metadata only Time-dependent diffeomorphisms as quantum canonical transformations and the time-dependent harmonic oscillator(Iop Publishing Ltd, 1998) N/A; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231Quantum canonical transformations corresponding to time-dependent diffeomorphisms of the configuration space are studied. A special class of these transformations which correspond to time-dependent dilatations is used to identify a previously unknown class of exactly solvable time-dependent harmonic oscillators. The Caldirola-Kanai oscillator belongs to this class. For a general time-dependent harmonic oscillator, it is shown that choosing the dilatation parameter to satisfy the classical equation of motion, one obtains the solution of the Schrodinger equation. A simple generalization of this result leads to the reduction of the Schrodinger equation to a second-order ordinary differential equation whose special case is the auxiliary equation of the Lewis-Riesenfeld invariant theory. The time-evolution operator is expressed in terms of a positive red solution of this equation in a closed form, and the time-dependent position and momentum operators are calculated.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 Asymptotic behavior of the irrational factor(Springer, 2008) Ledoan, A. H.; Zaharescu, Alexandru; Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803We study the irrational factor function I(n) introduced by Atanassov and defined by I(n) = Pi(k)(k=1)p(v)(1/alpha v), where n = Pi(k)(v=1) p(v)(alpha v) is the prime factorization of n. We show that the sequence {G(n)/n}(n >= 1), where G(n) = Pi(n)(v=1) I(v)(1/n), is covergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I(n).Publication Metadata only Performance evaluation of an exact method for the obstacle neutralization problem(IEOM Society, 2016) Alkaya, Ali Fuat; Algin, Ramazan; Oz, Dindar; Aksakalli, Vural; Department of Mathematics; Ceyhan, Elvan; Faculty Member; Department of Mathematics; College of Sciences; N/AThe Obstacle Neutralization Problem (ONP) is an NP-Hard path planning problem wherein an agent needs to swiftly navigate from a given start location to a target location through an arrangement of disc-shaped obstacles on the plane. The agent has a limited neutralization capability in the sense that it can neutralize an obstacle after which it can safely traverse through. A neutralization can only be performed at a cost, which is added to the overall traversal length. The goal is to find the optimal neutralization sequence that minimizes the agent's total traversal length. In this study, we compare the performance of a recently proposed exact algorithm for ONP against a conventional solution obtained via an integer programming formulation. This exact algorithm consists of two phases. In Phase I, an effective and fast algorithm is used to obtain a suboptimal solution. In the Phase II, a k-th shortest path algorithm is used to close any gaps. The integer programming formulation is solved via the popular SCIP solver. We present computational experiments conducted on synthetic problem instances on a discrete plane with varying resolutions. Our results indicate that the exact algorithm provides an almost 10-fold improvement in execution time when compared against the integer programming approach.Publication Metadata only On sums over the mobius function and discrepancy of fractions(Academic Press Inc Elsevier Science, 2013) Department of Mathematics; N/A; Alkan, Emre; Göral, Haydar; Faculty Member; Master Student; Department of Mathematics; College of Sciences; Graduate School of Sciences and Engineering; 32803; 252019We obtain quantitative upper bounds on partial sums of the Mobius function over semigroups of integers in an arithmetic progression. Exploiting, the cancellation of such sums, we deduce upper bounds for the discrepancy of fractions in the unit interval [0, 1] whose denominators satisfy the same restrictions. In particular, the uniform distribution and approximation of discrete weighted averages over such fractions are established as a consequence.