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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 A characterization of the closed unital ideals of the Fourier-Stieltjes algebra B(G) of a locally compact amenable group G(Elsevier, 2003) Department of Mathematics; Ülger, Ali; Faculty Member; Department of Mathematics; College of Sciences; N/ALet G be a locally compact amenable group, B(G) its Fourier–Stieltjes algebra and I be a closed ideal of it. In this paper we prove the following result: The ideal I has a unit element iff it is principal. This is the noncommutative version of the Glicksberg–Host–Parreau Theorem. The paper also contains an abstract version of this theorem.Publication Metadata only A note on the algebra of p-adic multi-zeta values(International Press of Boston, 2015) Department of Mathematics; Ünver, Sinan; Faculty Member; Department of Mathematics; College of Sciences; 177871We prove that the algebra of p-adic multi-zeta values, as defined in [4] or [2], are contained in another algebra which is defined explicitly in terms of series. The main idea is to truncate certain series, expand them in terms of series all of which are divergent except one, and then take the limit of the convergent one. The main result is Theorem 3.12.Publication Open Access A stochastic representation for mean curvature type geometric flows(Institute of Mathematical Statistics (IMS), 2003) Touzi, N.; Department of Mathematics; Soner, Halil Mete; Faculty Member; Department of Mathematics; College of Administrative Sciences and EconomicsA smooth solution {Gamma(t)}(tis an element of[0,T]) subset of R-d of a parabolic geometric flow is characterized as the reachability set of a stochastic target problem. In this control problem the controller tries to steer the state process into a given deterministic set T with probability one. The reachability set, V(t), for the target problem is the set of all initial data x from which the state process X-X(v)(t) is an element of T for some control process v. This representation is proved by studying the squared distance function to Gamma(t). For the codimension k mean curvature flow, the state process is dX(t) = root2P dW(t), where W(t) is a d-dimensional Brownian motion, and the control P is any projection matrix onto a (d - k)-dimensional plane. Smooth solutions of the inverse mean curvature flow and a discussion of non smooth solutions are also given.Publication Metadata only A stochastic representation for the level set equations(Taylor & Francis Inc, 2002) Touzi, Nizar; Department of Mathematics; Soner, Halil Mete; Faculty Member; Department of Mathematics; College of Sciences; N/AA Feynman-Kac representation is proved for geometric partial differential equations. This representation is in terms of a stochastic target problem. In this problem the controller tries to steer a controlled process into a given target by judicial choices of controls. The sublevel sets of the unique level set solution of the geometric equation is shown to coincide with the reachability sets of the target problem whose target is the sublevel set of the final data.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 Additive polylogarithms and their functional equations(Springer Heidelberg, 2010) Department of Mathematics; Ünver, Sinan; Faculty Member; Department of Mathematics; College of Sciences; 177871Let k[epsilon](2) := k[epsilon]/(epsilon(2)). The single valued real analytic n-polylogarithm L-n : C -> R is fundamental in the study of weight n motivic cohomology over a field k, of characteristic 0. In this paper, we extend the construction in Unver (Algebra Number Theory 3:1-34, 2009) to define additive n-polylogarithms li(n):k[epsilon](2) -> k and prove that they satisfy functional equations analogous to those of Ln. Under a mild hypothesis, we show that these functions descend to an analog of the nth Bloch group B'(n)(k[epsilon](2)) defined by Goncharov (Adv Math 114:197-318, 1995). We hope that these functions will be useful in the study of weight n motivic cohomology over k[epsilon](2).Publication Metadata only An abstract form of a theorem of Helson and applications to sets of synthesis and sets of uniqueness(Academic Press Inc Elsevier Science, 2010) Department of Mathematics; Ülger, Ali; Faculty Member; Department of Mathematics; College of Sciences; N/ALet E be a compact perfect subset of the real line R such that the restriction of the Fourier transform a bar right arrow (a) over cap vertical bar(E) from L(1)(R) into C(E) is onto. Helson proved that then, for mu is an element of M(E), lim(vertical bar y vertical bar ->infinity vertical bar) (u) over cap (y)vertical bar = 0 is possible only if mu = 0. In this paper we present an abstract version of this theorem of Helson and provide some applications of it to the study of sets of spectral synthesis and sets of uniqueness.Publication Metadata only Approximation by special values of harmonic zeta function and log-sine integrals(Int Press Boston, Inc, 2013) Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803Motivated by applications of log-sine integrals to a wide range of mathematical and physical problems, it is shown that real numbers and certain types of log-sine integrals can be strongly approximated by linear combinations of special values of the harmonic zeta function with the property that the coefficients belonging to these combinations turn out to be universal in the sense of being independent of special values. The approximation of real numbers by combinations of special values is reminiscent of the classical Diophantine approximation of Liouville numbers by rationals. Moreover, explicit representations of some specific log-sine integrals are obtained in terms of special values of the harmonic zeta function and the Riemann zeta function through a study of Fourier series involving harmonic numbers. In particular, special values of the harmonic zeta function and the less studied odd harmonic zeta function are expressed in terms of log-sine integrals over [0, 2 pi] and [0, pi].Publication Open Access Asymptotic H-Plateau problem in H-3(Mathematical Sciences Publishers (MSP), 2016) Department of Mathematics; Coşkunüzer, Barış; Faculty Member; Department of Mathematics; College of SciencesWe show that for any Jordan curve Gamma in S-infinity(2) (H-3) with at least one smooth point, there exists an embedded H-plane P-H in H-3 with partial derivative P-infinity(H) = Gamma for any H is an element of [0, 1).