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Publication Open Access A note on weakly compact homomorphisms between uniform algebras(Polish Academy of Sciences, 1997) Department of Mathematics; Ülger, Ali; Faculty Member; Department of Mathematics; College of SciencesPublication Open Access On the number of solutions to the asymptotic plateau problem(Gökova Geometry Topology (GGT) Conferences, 2011) Department of Mathematics; Coşkunüzer, Barış; Faculty Member; Department of Mathematics; College of SciencesBy using a simple topological argument, we show that the space of closed, orientable, codimension-1 submanifolds of Sn−1 1 (Hn) which bound a unique absolutely area minimizing hypersurface in Hn is dense in the space of closed, orientable, codimension-1 submanifolds of Sn−1 1 (Hn). In particular, in dimension 3, we prove that the set of simple closed curves in S2 1(H3) bounding a unique absolutely area minimizing surface in H3 is not only dense, but also a countable intersection of open dense subsets of the space of simple closed curves in S2 1(H3) with C0 topology. We also show that the same is true for least area planes in H3. Moreover, we give some non-uniqueness results in dimension 3.Publication Open Access Quantum mechanics of a photon(American Institute of Physics (AIP) Publishing, 2017) Department of Physics; Department of Mathematics; Babaei, Hassan; Mostafazadeh, Ali; Faculty Member; Department of Physics; Department of Mathematics; Graduate School of Sciences and Engineering; N/A; 4231A first-quantized free photon is a complex massless vector field A = (A(mu)) whose field strength satisfies Maxwell's equations in vacuum. We construct the Hilbert space H of the photon by endowing the vector space of the fields A in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in H, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated symmetry axis and show that each choice of this axis specifies a particular position operator, a corresponding position basis, and a position representation of the quantum mechanics of a photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and give an explicit formula for the probability density of the spatial localization of the photon.Publication Open Access The weak phillips property(Institute of Mathematics, Polish Academy of Sciences, 2001) Department of Mathematics; Ülger, Ali; Faculty Member; Department of Mathematics; College of SciencesLet X be a Banach space. If the natural projection p : X∗∗∗ → X∗ is sequentially weak∗ -weak continuous then the space X is said to have the weak Phillips property. We present several characterizations of the spaces having this property and study its relationships to other Banach space properties, especially the Grothendieck property.Publication Open Access Toward an abstract cogalois theory (I): Kneser and Cogalois groups of cocycles(The Institute of Mathematics of the Romanian Academy, 2004) Basarab, Şerban A.; Department of Mathematics; Albu, Toma; Faculty Member; Department of Mathematics; College of SciencesThis is the first part of a series of papers which aim to develop an abstract group theoretic framework for the Cogalois Theory of field extensions.