93,511 research outputs found

    Linear relations with conjugates of a Salem number

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    In this paper we consider linear relations with conjugates of a Salem number α\alpha. We show that every such a relation arises from a linear relation between conjugates of the corresponding totally real algebraic integer α+1/α\alpha+1/\alpha. It is also shown that the smallest degree of a Salem number with a nontrivial relation between its conjugates is 88, whereas the smallest length of a nontrivial linear relation between the conjugates of a Salem number is 66.Comment: v1, 12 page

    Natural PDE's of Linear Fractional Weingarten surfaces in Euclidean Space

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    We prove that the natural principal parameters on a given Weingarten surface are also natural principal parameters for the parallel surfaces of the given one. As a consequence of this result we obtain that the natural PDE of any Weingarten surface is the natural PDE of its parallel surfaces. We show that the linear fractional Weingarten surfaces are exactly the surfaces satisfying a linear relation between their three curvatures. Our main result is classification of the natural PDE's of Weingarten surfaces with linear relation between their curvatures.Comment: 16 page

    The distribution of supermassive black holes in the nuclei of nearby galaxies

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    The growth of supermassive black holes by merging and accretion in hierarchical models of galaxy formation is studied by means of Monte Carlo simulations. A tight linear relation between masses of black holes and masses of bulges arises if if the mass accreted by supermassive black holes scales linearly with the mass forming stars and if the redshift evolution of mass accretion tracks closely that of star formation. Differences in redshift evolution between black hole accretion and star formation introduce considerable scatter in this relation. A non-linear relation between black hole accretion and star formation results in a non-linear relation between masses of remnant black holes and masses of bulges. The relation of black hole mass to bulge luminosity obseved in nearby galaxies and its scatter are reproduced reasonably well by models in which black hole accretion and star formation are linearly related but do not track each other in redshift. This suggests that a common mechanism determines the efficiency for black hole accretion and the efficiency for star formation, especially for bright bulges.Comment: 6 pages, 3 figures, submitted to MNRA

    A family of linearizable recurrences with the Laurent property

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    We consider a family of non-linear recurrences with the Laurent property. Although these recurrences are not generated by mutations in a cluster algebra, they fit within the broader framework of Laurent phenomenon algebras, as introduced recently by Lam and Pylyavskyy. Furthermore, each member of this family is shown to be linearizable in two different ways, in the sense that its iterates satisfy both a linear relation with constant coefficients and a linear relation with periodic coefficients. Associated monodromy matrices and first integrals are constructed, and the connection with the dressing chain for Schrödinger operators is also explained

    A New Relation between post and pre-optimal measurement states

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    When an optimal measurement is made on a qubit and what we call an Unbiased Mixture of the resulting ensembles is taken, then the post measurement density matrix is shown to be related to the pre-measurement density matrix through a simple and linear relation. It is shown that such a relation holds only when the measurements are made in Mutually Unbiased Bases- MUB. For Spin-1/2 it is also shown explicitly that non-orthogonal measurements fail to give such a linear relation no matter how the ensembles are mixed. The result has been proved to be true for arbitrary quantum mechanical systems of finite dimensional Hilbert spaces. The result is true irrespective of whether the initial state is pure or mixed.Comment: 4 pages in REVTE

    On the Estimation of the Linear Relation when the Error Variances are known

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    The present article considers the problem of consistent estimation in measurement error models. A linear relation with not necessarily normally distributed measurement errors is considered. Three possible estimators which are constructed as different combinations of the estimators arising from direct and inverse regression are considered. The efficiency properties of these three estimators are derived and analyzed. The effect of non-normally distributed measurement errors is analyzed. A Monte-Carlo experiment is conducted to study the performance of these estimators in finite samples and the effect of a non-normal distribution of the measurement errors

    Scaling of the Strain Hardening Modulus of Glassy Polymers with the Flow Stress

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    In a recent letter, Govaert et al. examined the relationship between strain hardening modulus GrG_r and flow stress σflow\sigma_{flow} for five different glassy polymers. In each case, results for GrG_r at different strain rates or different temperatures were linearly related to the flow stress. They suggested that this linear relation was inconsistent with simulations. Data from previous publications and new results are presented to show that simulations also yield a linear relation between modulus and flow stress. Possible explanations for the change in the ratio of modulus to flow stress with temperature and strain rate are discussed.Comment: 8 pages, 2 figures: clarified arguments on linear proportionality. Accepted for publication in J. Poly. Sci Part B - Polym. Phy

    On a generalization of restricted sum formula for multiple zeta values and finite multiple zeta values

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    We prove a new linear relation for multiple zeta values. This is a natural generalization of the restricted sum formula proved by Eie, Liaw and Ong. We also present an analogous result for finite multiple zeta values
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