Publications without Fulltext
Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/3
Browse
448 results
Filters
Advanced Search
Filter by
Settings
Search Results
Item Metadata only A characterization of quasipositive two-bridge knots(World Scientific Publ Co Pte Ltd, 2024) 0000-0002-9758-1045; Orevkov, Stepan; Department of Mathematics; Özbağcı, Burak; Faculty Member; College of Sciences; 29746We prove a simple necessary and sufficient condition for a two-bridge knot K(p,q) to be quasipositive, based on the continued fraction expansion of p/q. As an application, coupled with some classification results in contact and symplectic topology, we give a new proof of the fact that smoothly slice two-bridge knots are non-quasipositive. Another proof of this fact using methods within the scope of knot theory is presented in Appendix A, by Stepan Orevkov.Item Metadata only Generalizations of planar contact manifolds to higher dimensions(International Press, Inc., 2023) 0000-0002-9758-1045; Acu, Bahar; Etnyre, John B.; Department of Mathematics; Özbağcı, Burak; Faculty Member; College of Sciences; 29746Iterated planar contact manifolds are a generalization of three dimensional planar contact manifolds to higher dimensions. We study some basic topological properties of iterated planar contact manifolds and discuss several examples and constructions showing that many contact manifolds are iterated planar. We also observe that for any odd integer m > 3, any finitely presented group can be re-alized as the fundamental group of some iterated planar contact manifold of dimension m. Moreover, we introduce another generalization of three dimensional planar contact manifolds that we call projective. Finally, building symplectic cobordisms via open books, we show that some projective contact manifolds admit explicit symplectic caps. © 2023, International Press, Inc.. All rights reserved.Item Metadata only General method for solving electromagnetic radiation problems in an arbitrary linear medium(Amer Physical Soc, 2023) 0000-0002-0739-4060; Loran, Farhang; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; College of Sciences; 4231Numerical transfer matrices have been widely used in the study of wave propagation and scattering. These may be viewed as discretizations of a recently introduced fundamental notion of transfer matrix which admits a representation in terms of the evolution operator for an effective nonunitary quantum system. We use the fundamental transfer matrix to develop a general method for the solution of the problem of radiation of an oscillating source in an arbitrary, possibly nonhomogeneous, anisotropic, and active or lossy linear medium. This allows us to obtain an analytic solution to this problem for an oscillating source located in the vicinity of a planar collection of possibly anisotropic and active or lossy point scatterers such as those modeling a two-dimensional photonic crystal.Item Metadata only Fundamental transfer matrix for electromagnetic waves, scattering by a planar collection of point scatterers, and anti-PT-symmetry(Amer Physical Soc, 2023) 0000-0002-0739-4060; Loran, Farhang; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; College of Sciences; 4231We develop a fundamental transfer-matrix formulation of the scattering of electromagnetic (EM) waves that incorporates the contribution of the evanescent waves and applies to general stationary linear media which need not be isotropic, homogenous, or passive. Unlike the traditional transfer matrices whose definition involves slicing the medium, the fundamental transfer matrix is a linear operator acting in an infinite-dimensional function space. It is given in terms of the evolution operator for a nonunitary quantum system and has the benefit of allowing for analytic calculations. In this respect it is the only available alternative to the standard Green'sfunction approaches to EM scattering. We use it to offer an exact solution of the outstanding EM scattering problem for an arbitrary finite collection of possibly anisotropic nonmagnetic point scatterers lying on a plane. In particular, we provide a comprehensive treatment of doublets consisting of pairs of isotropic point scatterers and study their spectral singularities. We show that identical and PT-symmetric doublets do not admit spectral singularities and cannot function as a laser unless the real part of their permittivity equals that of the vacuum. This restriction does not apply to doublets displaying anti-PT-symmetry. We determine the lasing threshold for a generic anti-PT-symmetric doublet and show that it possesses a continuous lasing spectrum.Item Metadata only Broadband directional invisibility(AIP Publishing, 2023) 0000-0002-0739-4060; Loran, Farhang; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; College of Sciences; 4231The discovery of unidirectional invisibility and its broadband realization in optical media satisfying spatial Kramers-Kronig relations are important landmarks of non-Hermitian photonics. We offer a precise characterization of a higher-dimensional generalization of this effect and find sufficient conditions for its realization in the scattering of scalar waves in two and three dimensions and electromagnetic waves in three dimensions. More specifically, given a positive real number alpha and a continuum of unit vectors Omega, we provide explicit conditions on the interaction potential (or the permittivity and permeability tensors of the scattering medium in the case of electromagnetic scattering) under which it displays perfect (non-approximate) invisibility whenever the incident wavenumber k does not exceed alpha (i.e., k is an element of( 0 , alpha ]) and the direction of the incident wave vector ranges over Omega. A distinctive feature of our approach is that it allows for the construction of potentials and linear dielectric media that display perfect directional invisibility in a finite frequency domain.Item Metadata only An uncountable Moore-Schmidt theorem(Cambridge University Press, 2023) 0000-0002-1450-6569; Tao, Terence; Department of Mathematics; Jamneshan, Asgar; Faculty Member; College of Sciences; 332404We prove an extension of the Moore-Schmidt theorem on the triviality of the first cohomology class of cocycles for the action of an arbitrary discrete group on an arbitrary measure space and for cocycles with values in an arbitrary compact Hausdorff abelian group. The proof relies on a 'conditional' Pontryagin duality for spaces of abstract measurable maps. © The Author(s), 2022. Published by Cambridge University Press.Item Metadata only An uncountable Furstenberg-Zimmer structure theory(Cambridge University Press, 2023) 0000-0002-1450-6569; Tao, Terence; Department of Mathematics; Jamneshan, Asgar; Faculty Member; College of Sciences; 332404Furstenberg-Zimmer structure theory refers to the extension of the dichotomy between the compact and weakly mixing parts of a measure-preserving dynamical system and the algebraic and geometric descriptions of such parts to a conditional setting, where such dichotomy is established relative to a factor and conditional analogs of those algebraic and geometric descriptions are sought. Although the unconditional dichotomy and the characterizations are known for arbitrary systems, the relative situation is understood under certain countability and separability hypotheses on the underlying groups and spaces. The aim of this article is to remove these restrictions in the relative situation and establish a Furstenberg-Zimmer structure theory in full generality. As an independent byproduct, we establish a connection between the relative analysis of systems in ergodic theory and the internal logic in certain Boolean topoi. © The Author(s), 2022. Published by Cambridge University Press.Item Metadata only Foundational aspects of uncountable measure theory: gelfand duality, riesz representation, canonical models, and canonical disintegration(Institute of Mathematics. Polish Academy of Sciences, 2023) 0000-0002-1450-6569; Tao, Terence; Department of Mathematics; Jamneshan, Asgar; Faculty Member; College of Sciences; 332404We collect several foundational results regarding the interaction between locally compact spaces, probability spaces and probability algebras, and commutative C∗-algebras and von Neumann algebras equipped with traces, in the “uncountable” setting in which no separability, metrizability, or standard Borel hypotheses are placed on these spaces and algebras. In particular, we review the Gelfand dualities and Riesz representation theorems available in this setting. We also present a canonical model that represents probability algebras as compact Hausdorff probability spaces in a completely functorial fashion, and apply this model to obtain a canonical disintegration theorem and to readily construct various product measures. These tools are useful in applications to “uncountable” ergodic theory (as demonstrated by the authors and others). © 2023 Institute of Mathematics. Polish Academy of Sciences. All rights reserved.Item Metadata only The inverse theorem for the U3 gowers uniformity norm on arbitrary finite abelian groups: fourier-analytic and ergodic approaches(Alliance of Diamond OA Journals, 2023) 0000-0002-1450-6569; Tao, Terence; Department of Mathematics; Jamneshan, Asgar; Faculty Member; College of Sciences; 332404We state and prove a quantitative inverse theorem for the Gowers uniformity norm U3(G) on an arbitrary finite abelian group G; the cases when G was of odd order or a vector space over F2 had previously been established by Green and the second author and by Samorodnitsky respectively by Fourier-analytic methods, which we also employ here. We also prove a qualitative version of this inverse theorem using a structure theorem of Host–Kra type for ergodic Zω-actions of order 2 on probability spaces established recently by Shalom and the authors. © 2023 Asgar Jamneshan, and Terence TaoItem Metadata only On the local well-posedness of the 1D Green-Naghdi system on Sobolev spaces(Wiley-V C H Verlag Gmbh, 2023) 0000-0001-9626-5000; Department of Mathematics; İnci, Hasan; Faculty Member; College of Sciences; 274184In this paper, we consider the local well-posedness of the 1D Green-Naghdi system. This system describes the evolution of water waves over an uneven bottom in the shallow water regime in terms of the water depth h and the horizontal velocity u. Using a Lagrangian formulation of this system on a Sobolev-type diffeomorphism group, we prove local well-posedness for (h,u) in the Sobolev space ([1-xi]+Hs(R))xHs+1(R),s>1/2, where xi : R -> R is the parameterization of the bottom and where we assume that the water surface has an equilibrium at height 1. This improves the present local well-posedness range by one degree.