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Unit Disk (unit + disk)
Selected AbstractsExploratory orientation data analysis with , sectionsJOURNAL OF APPLIED CRYSTALLOGRAPHY, Issue 5 2004K. Gerald Van Den Boogaart Since the domain of crystallographic orientations is three-dimensional and spherical, insightful visualization of them or visualization of related probability density functions requires (i) exploitation of the effect of a given orientation on the crystallographic axes, (ii) consideration of spherical means of the orientation probability density function, in particular with respect to one-dimensional totally geodesic submanifolds, and (iii) application of projections from the two-dimensional unit sphere onto the unit disk . The familiar crystallographic `pole figures' are actually mean values of the spherical Radon transform. The mathematical Radon transform associates a real-valued function f defined on a sphere with its mean values along one-dimensional circles with centre , the origin of the coordinate system, and spanned by two unit vectors. The family of views suggested here defines , sections in terms of simultaneous orientational relationships of two different crystal axes with two different specimen directions, such that their superposition yields a user-specified pole probability density function. Thus, the spherical averaging and the spherical projection onto the unit disk determine the distortion of the display. Commonly, spherical projections preserving either volume or angle are favoured. This rich family displays f completely, i.e. if f is given or can be determined unambiguously, then it is uniquely represented by several subsets of these views. A computer code enables the user to specify and control interactively the display of linked views, which is comprehensible as the user is in control of the display. [source] The ice-fishing problem: the fundamental sloshing frequency versus geometry of holesMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2004Vladimir Kozlov Abstract We study an eignevalue problem with a spectral parameter in a boundary condition. This problem for the Laplace equation is relevant to sloshing frequencies that describe free oscillations of an inviscid, incompressible, heavy fluid in a half-space covered by a rigid dock with some apertures (an ice sheet with fishing holes). The dependence of the fundamental eigenvalue on holes' geometry is investigated. We give conditions on a plane region guaranteeing that the fundamental eigenvalue corresponding to this region is larger than the fundamental eigenvalue corresponding to a single circular hole. Examples of regions satisfying these conditions and having the same area as the unit disk are given. New results are also obtained for the problem with a single circular hole. On the other hand, we construct regions for which the fundamental eigenfrequency is larger than the similar frequency for the circular hole of the same area and even as large as one wishes. In the latter examples, the hole regions are either not connected or bounded by a rather complicated curves. Copyright © 2004 John Wiley & Sons, Ltd. [source] Carathéodory,Julia type conditions and symmetries of boundary asymptotics for analytic functions on the unit diskMATHEMATISCHE NACHRICHTEN, Issue 11 2009Vladimir Bolotnikov Abstract It is shown that the following conditions are equivalent for the generalized Schur class functions at a boundary point t0 , ,,: 1) Carathéodory,Julia type condition of order n; 2) agreeing of asymptotics of the original function from inside and of its continuation by reflection from outside of the unit disk ,, up to order 2n + 1; 3) t0 -isometry of the coefficients ofthe boundary asymptotics; 4) a certain structured matrix , constructed from these coefficients being Hermitian. It is also shown that for an arbitrary analytic function, properties 2), 3), 4) are still equivalent to each other and imply 1) (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] On a class of holomorphic functions representable by Carleman formulas in the disk from their values on the arc of the circleMATHEMATISCHE NACHRICHTEN, Issue 1-2 2007Lev Aizenberg Abstract Let D be a unit disk andM be an open arc of the unit circle whose Lebesgue measure satisfies 0 < l (M) < 2,. Our first result characterizes the restriction of the holomorphic functions f , ,(D), which are in the Hardy class ,1 near the arcM and are denoted by ,, ,1M(,,), to the open arcM. This result is a direct consequence of the complete description of the space of holomorphic functions in the unit disk which are represented by the Carleman formulas on the open arc M. As an application of the above characterization, we present an extension theorem for a function f , L1(M) from any symmetric sub-arc L , M of the unit circle, such that , M, to a function f , ,, ,1L(,,). (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Limits of zeros of orthogonal polynomials on the circleMATHEMATISCHE NACHRICHTEN, Issue 12-13 2005Barry Simon Abstract We prove that there is a universal measure on the unit circle such that any probability measure on the unit disk is the limit distribution of some subsequence of the corresponding orthogonal polynomials. This follows from an extension of a result of Alfaro and Vigil (which answered a question of P. Turán): namely, for n < N , one can freely prescribe the n -th polynomial and N , n zeros of the N -th one. We shall also describe all possible limit sets of zeros within the unit disk. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Shift operators contained in contractions and pseudocontinuable matrixvalued Schur functionsMATHEMATISCHE NACHRICHTEN, Issue 7-8 2005S. S. Boiko Abstract We show that the pseudocontinuability of a matrix-valued Schur function , in the unit disk is completely determined by the properties of the maximal shift and maximal coshift contained in a corresponding contractive operator T which has the characteristic operator function ,. [source] | ||||||||||