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Journal articles on the topic 'Birkhoff'

Author: Grafiati

Published: 4 June 2021

Last updated: 19 February 2022

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1

Chen, Jin-Yue, and Yi Zhang. "Time-Scale Version of Generalized Birkhoffian Mechanics and Its Symmetries and Conserved Quantities of Noether Type." Advances in Mathematical Physics 2021 (April 26, 2021): 1–9. http://dx.doi.org/10.1155/2021/9982975.

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The time-scale version of Noether symmetry and conservation laws for three Birkhoffian mechanics, namely, nonshifted Birkhoffian systems, nonshifted generalized Birkhoffian systems, and nonshitfed constrained Birkhoffian systems, are studied. Firstly, on the basis of the nonshifted Pfaff-Birkhoff principle on time scales, Birkhoff’s equations for nonshifted variables are deduced; then, Noether’s quasi-symmetry for the nonshifted Birkhoffian system is proved and time-scale conserved quantity is presented. Secondly, the nonshifted generalized Pfaff-Birkhoff principle on time scales is proposed, the generalized Birkhoff’s equations for nonshifted variables are derived, and Noether’s symmetry for the nonshifted generalized Birkhoffian system is established. Finally, for the nonshifted constrained Birkhoffian system, Noether’s symmetry and time-scale conserved quantity are proposed and proved. The validity of the result is proved by examples.

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2

Gehrke, Mai, and Michael Pinsker. "Uniform Birkhoff." Journal of Pure and Applied Algebra 222, no. 5 (May 2018): 1242–50. http://dx.doi.org/10.1016/j.jpaa.2017.06.016.

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3

Bodirsky, Manuel, and Michael Pinsker. "Topological Birkhoff." Transactions of the American Mathematical Society 367, no. 4 (August 8, 2014): 2527–49. http://dx.doi.org/10.1090/s0002-9947-2014-05975-8.

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4

Biswas, HR, and MS Islam. "Ergodic theory of one dimensional Map." Bangladesh Journal of Scientific and Industrial Research 47, no. 3 (December 21, 2012): 321–26. http://dx.doi.org/10.3329/bjsir.v47i3.13067.

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In this paper we study one dimensional linear and non-linear maps and its dynamical behavior. We study measure theoretical dynamical behavior of the maps. We study ergodic measure and Birkhoff ergodic theorem. Also, we study some problems using Birkhoff's ergodic theorem. DOI: http://dx.doi.org/10.3329/bjsir.v47i3.13067 Bangladesh J. Sci. Ind. Res. 47(3), 321-326 2012

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5

Kurnianto, Arik. "Analisis Layout Surat Kabar berdasarkan Prinsip-Prinsip Desain melalui Metode Estetika Birkhoff." Humaniora 4, no. 2 (October 31, 2013): 986. http://dx.doi.org/10.21512/humaniora.v4i2.3540.

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This study has primary focus to analyze the aesthetic layout of a newspaper with a mathematical approach, which refers to some methods of Birkhoff’s aesthetic measurement developed by David Ngo Chek Ling, et.al. Ngo developed a method of measuring the aesthetic layout that can be used for various design purposes, especially for a computer interface design. Measurement method that he developed refers to the principles of design aesthetics, such as: the principle of balance, the principle of sequence, the principle of unity, and the principle of equilibrium. Birkhoff aesthetic measurement method was used as final analysis to determine the value of aesthetic layout. In this study, some national and local media were selected as samples to determine and assess the aesthetic layout. Based on data analysis, Birkhoff aesthetic measurement method can be applied to measure the aesthetic value of the newspaper layout mathematically.

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6

Buczolich, Zoltán, Balázs Maga, and Ryo Moore. "Generic Birkhoff spectra." Discrete & Continuous Dynamical Systems - A 40, no. 12 (2020): 6649–79. http://dx.doi.org/10.3934/dcds.2020131.

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7

Madden, James J. "Pierce-Birkhoff rings." Archiv der Mathematik 53, no. 6 (December 1989): 565–70. http://dx.doi.org/10.1007/bf01199816.

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8

Möller, Manfred. "Expansion theorems for Birkhoff-regular differential-boundary operators." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 107, no. 3-4 (1987): 349–74. http://dx.doi.org/10.1017/s0308210500031218.

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SynopsisIn this paper we consider differential-boundary operators T over a finite interval depending on a complex parameter. A differential-boundary operator admits boundary conditions in the differential part. The boundary part contains multipoint boundary conditions and integral conditions. For Birkhoff-regular boundary conditions we prove that every Lp -function is expansible into a series with respect to the eigenfunctions and the associated functions of the differential-boundary operator. Here the Birkhoff-regularity only depends on the boundary conditions at the endpoints of the interval, i.e. T is Birkhoff-regular if and only if T0 is Birkhoff-regular where T0 arises from T by omitting the boundary part in the differential equations, the interior point boundary conditions and the integral condition.

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9

WÓJCIK, PAWEŁ. "BIRKHOFF ORTHOGONALITY IN CLASSICAL -IDEALS." Journal of the Australian Mathematical Society 103, no. 2 (November 8, 2016): 279–88. http://dx.doi.org/10.1017/s1446788716000537.

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The Birkhoff orthogonality has been recently intensively studied in connection with the geometry of Banach spaces and operator theory. The main aim of this paper is to characterize the Birkhoff orthogonality in ${\mathcal{L}}(X;Y)$ under the assumption that ${\mathcal{K}}(X;Y)$ is an $M$-ideal in ${\mathcal{L}}(X;Y)$. Moreover, we survey the known results, as well as giving some new and more general ones. Furthermore, we characterize an approximate Birkhoff orthogonality in ${\mathcal{K}}(X;Y)$.

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10

Snamina, Mateusz, and Emil J. Zak. "Dynamical Semigroups in the Birkhoff Polytope of Order 3 as a Tool for Analysis of Quantum Channels." Open Systems & Information Dynamics 27, no. 01 (March 2020): 2050001. http://dx.doi.org/10.1142/s1230161220500018.

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In the present paper we show a link between bistochastic quantum channels and classical maps. The primary goal of this work is to analyse the multiplicative structure of the Birkhoff polytope of order 3 (the simplest nontrivial case). A suitable complex parametrization of the Birkhoff polytope is proposed, which reveals several its symmetries and characteristics, in particular: (i) the structure of Markov semigroups inside the Birkhoff polytope, (ii) the relation between the set of Markov time evolutions, the set of positive definite matrices and the set of divisible matrices. A condition for Markov time evolution of semigroups in the set of symmetric bistochastic matrices is derived, which leads to an universal conserved quantity for all Markov evolutions. Finally, the complex parametrization is extended to the Birkhoff polytope of order 4.

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11

Hariyanto, Susilo, Titi Udjiani, Muhammad Rafid Fadil, and Yuri C. Sagala. "Finding Minimum Distance on Birkhoff-James Orthogonality in Banach Space." EKSAKTA: Journal of Sciences and Data Analysis 20, no. 2 (July 20, 2020): 124–28. http://dx.doi.org/10.20885/eksakta.vol1.iss2.art5.

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In this paper, we define orthogonality concept on Banach space. This orthogonality is called Birkhoff-James orthogonality. In this paper, we will discuss some new problem about the correlation between orthogonality on Hilbert space and Birkhoff-James orthogonality. Correlation between those two can be investigated by observing the correlation between Hilbert space and Banach space with particular norm. Further, we discuss about the correlation between minimum distance in Banach space with Birkhoff-James orthogonality, by generalized the concept of finding the minimum distance in Hilbert space.

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12

Kar, S., S. Purkait, and R. Sarkar. "Birkhoff center of c-semiring." Asian-European Journal of Mathematics 12, no. 01 (February 2019): 1950003. http://dx.doi.org/10.1142/s1793557119500037.

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In this paper, we introduce the concept of Birkhoff center of a [Formula: see text]-semiring and provide some characterizations of this center. Finally, we prove that the Birkhoff center of a [Formula: see text]-semiring forms a distributive lattice.

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13

Chierchia, Luigi, and Gabriella Pinzari. "Planetary Birkhoff normal forms." Journal of Modern Dynamics 5, no. 4 (2011): 623–64. http://dx.doi.org/10.3934/jmd.2011.5.623.

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14

FAN, AI-HUA, and JÖRG SCHMELING. "On fast Birkhoff averaging." Mathematical Proceedings of the Cambridge Philosophical Society 135, no. 3 (November 2003): 443–67. http://dx.doi.org/10.1017/s0305004103006819.

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15

Tabachnikov, Serge. "Birkhoff billiards are insecure." Discrete & Continuous Dynamical Systems - A 23, no. 3 (2009): 1035–40. http://dx.doi.org/10.3934/dcds.2009.23.1035.

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16

Schneider, Friedrich Martin. "A uniform Birkhoff theorem." Algebra universalis 78, no. 3 (October 17, 2017): 337–54. http://dx.doi.org/10.1007/s00012-017-0460-1.

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17

Jordan, Thomas, Vincent Naudot, and Todd Young. "Higher order Birkhoff averages." Dynamical Systems 24, no. 3 (August 17, 2009): 299–313. http://dx.doi.org/10.1080/14689360802676269.

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18

Deser, S., and J. Franklin. "Birkhoff for Lovelock redux." Classical and Quantum Gravity 22, no. 16 (July 26, 2005): L103—L106. http://dx.doi.org/10.1088/0264-9381/22/16/l03.

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19

Varma, A. K. "On Birkhoff quadrature formulas." Proceedings of the American Mathematical Society 97, no. 1 (January 1, 1986): 38. http://dx.doi.org/10.1090/s0002-9939-1986-0831383-9.

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20

Hack, F. J. "On bivariate Birkhoff interpolation." Journal of Approximation Theory 49, no. 1 (January 1987): 18–30. http://dx.doi.org/10.1016/0021-9045(87)90110-9.

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21

Goswami, Rituparno, and George F. R. Ellis. "Birkhoff theorem and matter." General Relativity and Gravitation 44, no. 8 (May 11, 2012): 2037–50. http://dx.doi.org/10.1007/s10714-012-1376-x.

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22

Kowalski, Klaus. "Reelle Hermite–Birkhoff-Interpolation." Results in Mathematics 62, no. 3-4 (September 8, 2012): 405–14. http://dx.doi.org/10.1007/s00025-012-0282-6.

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23

Anikin, A. Yu. "Quantum Birkhoff normal forms." Theoretical and Mathematical Physics 160, no. 3 (September 2009): 1274–91. http://dx.doi.org/10.1007/s11232-009-0115-2.

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24

Codescu, Mihai, and Daniel Găină. "Birkhoff Completeness in Institutions." Logica Universalis 2, no. 2 (September 24, 2008): 277–309. http://dx.doi.org/10.1007/s11787-008-0035-1.

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25

Kwon, Seok-Il. "A Didactical Analysis on the Birkhoff’s Axiomatic System." Journal of Curriculum and Evaluation 12, no. 3 (November 2009): 275–93. http://dx.doi.org/10.29221/jce.2009.12.3.275.

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26

OLVERA, ARTURO, and CARLES SIMÓ. "ELLIPTIC NON-BIRKHOFF PERIODIC ORBITS IN THE TWIST MAPS." International Journal of Bifurcation and Chaos 03, no. 01 (February 1993): 165–85. http://dx.doi.org/10.1142/s021812749300012x.

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We consider a perturbed twist map when the perturbation is big enough to destroy the invariant rotational curve (IRC) with a given irrational rotation number. Then an invariant Cantorian set appears. From another point of view, the destruction of the IRC is associated with the appearance of heteroclinic connections between hyperbolic periodic points. Furthermore the destruction of the IRC is also associated with the existence of non-Birkhoff orbits. In this paper we relate the different approaches. In order to explain the creation of non-Birkhoff orbits, we provide qualitative and quantitative models. We show the existence of elliptic non-Birkhoff periodic orbits for an open set of values of the perturbative parameter. The bifurcations giving rise to the elliptic non-Birkhoff orbits and other related bifurcations are analysed. In the last section, we show a celestial mechanics example displaying the described behavior.

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27

IOMMI, GODOFREDO, and THOMAS JORDAN. "Multifractal analysis of Birkhoff averages for countable Markov maps." Ergodic Theory and Dynamical Systems 35, no. 8 (August 25, 2015): 2559–86. http://dx.doi.org/10.1017/etds.2015.44.

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In this paper we prove a multifractal formalism of Birkhoff averages for interval maps with countably many branches. Furthermore, we prove that under certain assumptions the Birkhoff spectrum is real analytic. We also show that new phenomena occur; indeed, the spectrum can be constant or it can have points where it is not analytic. Conditions for these to happen are obtained. Applications of these results to number theory are also given. Finally, we compute the Hausdorff dimension of the set of points for which the Birkhoff average is infinite.

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28

Precupanu, T. "Duality Mapping and Birkhoff Orthogonality." Annals of the Alexandru Ioan Cuza University - Mathematics 59, no. 1 (January 1, 2013): 103–12. http://dx.doi.org/10.2478/v10157-012-0027-6.

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Abstract. In this note we establish some properties of Birkhoff orthogonality in terms of duality mapping. Particularly, using a slight extension of a Rockafellar result we obtain a new proof of a result earlier established by Blanco and Turnšek concerning the linear operators preserving Birkhoff orthogonality.

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29

Wang Chuan-Dong, Liu Shi-Xing, and Mei Feng-Xiang. "Generalized Pfaff-Birkhoff-d’Alembert principle and form invariance of generalized Birkhoff’s equations." Acta Physica Sinica 59, no. 12 (2010): 8322. http://dx.doi.org/10.7498/aps.59.8322.

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30

Le Calvez, P. "Propriétés des attracteurs de Birkhoff." Ergodic Theory and Dynamical Systems 8, no. 2 (June 1988): 241–310. http://dx.doi.org/10.1017/s0143385700004442.

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AbstractWe study dissipative twist maps of the annulus, following the ideas of G. D. Birkhoff explained in an article of 1932.In the first part, we give complete and rigorous proofs of the results of this article. We define the Birkhoff attractor of a dissipative twist map which has an attracting bounded annulus, we give its main properties and we define its upper and lower rotation numbers.In the second part we give further results on these sets, thus we show that they often coincide with the closure of a hyperbolic periodic point and that they can contain an infinite number of sinks. We also show that the Birkhoff attractors don't depend on a continuous way on the maps.

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31

MRAMOR, BLAŽ, and BOB RINK. "A dichotomy theorem for minimizers of monotone recurrence relations." Ergodic Theory and Dynamical Systems 35, no. 1 (September 27, 2013): 215–48. http://dx.doi.org/10.1017/etds.2013.47.

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AbstractVariational monotone recurrence relations arise in solid state physics as generalizations of the Frenkel–Kontorova model for a ferromagnetic crystal. For such problems, Aubry–Mather theory establishes the existence of ‘ground states’ or ‘global minimizers’ of arbitrary rotation number. A nearest neighbor crystal model is equivalent to a Hamiltonian twist map. In this case, the global minimizers have a special property: they can only cross once. As a non-trivial consequence, every one of them has the Birkhoff property. In crystals with a larger range of interaction and for higher order recurrence relations, the single crossing property does not hold and there can exist global minimizers that are not Birkhoff. In this paper we investigate the crossings of global minimizers. Under a strong twist condition, we prove the following dichotomy: they are either Birkhoff, and thus very regular, or extremely irregular and non-physical: they then grow exponentially and oscillate. For Birkhoff minimizers, we also prove certain strong ordering properties that are well known for twist maps.

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32

ITO, HIDEKAZU. "Birkhoff normalization and superintegrability of Hamiltonian systems." Ergodic Theory and Dynamical Systems 29, no. 6 (March 2, 2009): 1853–80. http://dx.doi.org/10.1017/s0143385708000965.

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AbstractWe study Birkhoff normalization in connection with superintegrability of ann-degree-of-freedom Hamiltonian systemXHwith holomorphic HamiltonianH. Without assuming any Poisson commuting relation among integrals, we prove that, if the system XHhasn+qholomorphic integrals near an equilibrium point of resonance degreeq≥0, there exists a holomorphic Birkhoff transformation φ such thatH∘φ becomes a holomorphic function ofn−qvariables and thatXH∘φcan be solved explicitly. Furthermore, the Birkhoff normal formH∘φ is determined uniquely, independently of the choice of φ, as convergent power series. We also show that the systemXHis superintegrable in the sense of Mischenko–Fomenko as well as Liouville-integrable near the equilibrium point.

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33

Kaluža, Matjaž. "Program for Birkhoff-Gustavson normal form for N degrees of freedom - BIRKHOFF 1.2." Computer Physics Communications 74, no. 3 (March 1993): 441–49. http://dx.doi.org/10.1016/0010-4655(93)90025-8.

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34

KOMURO, NAOTO, KICHI-SUKE SAITO, and RYOTARO TANAKA. "LEFT SYMMETRIC POINTS FOR BIRKHOFF ORTHOGONALITY IN THE PREDUALS OF VON NEUMANN ALGEBRAS." Bulletin of the Australian Mathematical Society 98, no. 3 (August 28, 2018): 494–501. http://dx.doi.org/10.1017/s0004972718000849.

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In this paper, we give a complete description of left symmetric points for Birkhoff orthogonality in the preduals of von Neumann algebras. As a consequence, except for $\mathbb{C}$, $\ell _{\infty }^{2}$ and $M_{2}(\mathbb{C})$, there are no von Neumann algebras whose preduals have nonzero left symmetric points for Birkhoff orthogonality.

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35

Wang, Ya-Nan, and Wen-Xin Qin. "A Necessary Condition of Nonminimal Aubry–Mather Sets for Monotone Recurrence Relations." International Journal of Bifurcation and Chaos 24, no. 01 (January 2014): 1450012. http://dx.doi.org/10.1142/s0218127414500126.

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In this paper, we show that a necessary condition for nonminimal Aubry–Mather sets of monotone recurrence relations is that the set of all Birkhoff minimizers with some irrational rotation number does not constitute a foliation, i.e. the gaps of the minimal Aubry–Mather set are not filled up with Birkhoff minimizers.

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36

Rouleux, Michel. "Semi-classical Integrability, Hyperbolic Flows and the Birkhoff Normal Form." Canadian Journal of Mathematics 56, no. 5 (October 1, 2004): 1034–67. http://dx.doi.org/10.4153/cjm-2004-047-6.

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AbstractWe prove that a Hamiltonianp∈C∞(T*Rn) is locally integrable near a non-degenerate critical point ρ0of the energy, provided that the fundamental matrix at ρ0has rationally independent eigenvalues, none purely imaginary. This is done by using Birkhoff normal forms, which turn out to be convergent in theC∞sense. We also give versions of the Lewis-Sternberg normal form near a hyperbolic fixed point of a canonical transformation. Then we investigate the complex case, showing that whenpis holomorphic near ρ0∈T*Cn, then Repbecomes integrable in the complex domain for real times, while the Birkhoff series and the Birkhoff transforms may not converge,i.e., pmay not be integrable. These normal forms also hold in the semi-classical frame.

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37

SAKAJO, TAKASHI. "Analytic continuation of the Birkhoff–Rott equation in complex-time domain." European Journal of Applied Mathematics 15, no. 1 (February 2004): 39–53. http://dx.doi.org/10.1017/s0956792503005230.

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A vortex sheet is a surface across which the velocity field of incompressible and inviscid flows has a jump discontinuity. Mathematical and numerical studies reveal that a two-dimensional vortex sheet, which is governed by the Birkhoff–Rott equation, acquires a singularity in finite time without forming rolling-up spiral. On the other hand, numerical computation of a regularized Birkhoff–Rott equation shows that the vortex sheet evolves into a rolling-up doubly branched spiral. Because of the finite-time singularity, it is impossible to regard the rolling-up spiral as a solution of the Birkhoff–Rott equation as long as time is real. However, it may be possible to analytically continue the equation to the spiral along a path to get around the singularity in complex-time plane. In the present article, we consider singularities in complex-time plane for the regularized Birkhoff–Rott equation by numerical means. Distribution of the complex singularities and their limiting behaviour indicate that it is absolutely impossible to perform analytic continuation in complex-time domain to the spiral solution. Furthermore, we propose a simple model of a doubly branched spiral and investigate it mathematically. The model is successful in approximating the rolling-up motion of the vortex sheet. Comparing the vortex-sheet motion with the model indicate that the doubly branched spiral with infinite windings at the centre could be a solution of the Birkhoff–Rott equation beyond the singularity time.

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38

Ojha, Bhuwan Prasad, and Prakash Muni Bajrayacharya. "Relation of Pythagorean and Isosceles Orthogonality with Best approximations in Normed Linear Space." Mathematics Education Forum Chitwan 4, no. 4 (November 15, 2019): 72–78. http://dx.doi.org/10.3126/mefc.v4i4.26360.

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In an arbitrary normed space, though the norm not necessarily coming from the inner product space, the notion of orthogonality may be introduced in various ways as suggested by the mathematicians like R.C. James, B.D. Roberts, G. Birkhoff and S.O. Carlsson. We aim to explore the application of orthogonality in normed linear spaces in the best approximation. Hence it has already been proved that Birkhoff orthogonality implies best approximation and best approximation implies Birkhoff orthogonality. Additionally, it has been proved that in the case of ε -orthogonality, ε -best approximation implies ε -orthogonality and vice-versa. In this article we established relation between Pythagorean orthogonality and best approximation as well as isosceles orthogonality and ε -best approximation in normed space.

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39

Françoise, Jean. "Birkhoff normal formsand analytic geometry." Banach Center Publications 39, no. 1 (1997): 49–56. http://dx.doi.org/10.4064/-39-1-49-56.

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40

Bojanov, Borislav. "B-splines with Birkhoff knots." Constructive Approximation 4, no. 1 (December 1988): 147–56. http://dx.doi.org/10.1007/bf02075455.

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41

Bhattacharyya, Tirthankar, and Priyanka Grover. "Characterization of Birkhoff–James orthogonality." Journal of Mathematical Analysis and Applications 407, no. 2 (November 2013): 350–58. http://dx.doi.org/10.1016/j.jmaa.2013.05.022.

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42

Hughes, Jesse. "Some Co-Birkhoff Type Theorems." Electronic Notes in Theoretical Computer Science 65, no. 1 (October 2002): 92–111. http://dx.doi.org/10.1016/s1571-0661(04)80361-x.

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43

Kappeler, T., B. Schaad, and P. Topalov. "mKdV and its Birkhoff coordinates." Physica D: Nonlinear Phenomena 237, no. 10-12 (July 2008): 1655–62. http://dx.doi.org/10.1016/j.physd.2008.03.018.

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44

Mccord, Christopher K., and Kenneth R. Meyer. "Two Conjectures of G.D. Birkhoff." International Astronomical Union Colloquium 172 (1999): 439–40. http://dx.doi.org/10.1017/s0252921100073061.

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The spatial (planar) three-body problem admits the ten (six) integrals of energy, center of mass, linear momentum and angular momentum. Fixing these integrals defines an eight (six) dimensional algebraic set called the integral manifold, 𝔐(c, h) (m(c, h)), which depends on the energy level h and the magnitude c of the angular momentum vector. The seven (five) dimensional reduced integral manifold, 𝔐R(c, h) (mR(c, h)), is the quotient space 𝔐(c, h)/SO2 (m(c, h)/SO2) where the SO2 action is rotation about the angular momentum vector. We want to determine how the geometry or topology of these sets depends on c and h. It turns out that there is one bifurcation parameter, ν = −c2h, and nme (six) special values of this parameter, νi, i = 1, …, 9.At each of the special values the geometric restrictions imposed by the integrals change, but one of these values, ν5, does not give rise to a change in the topology of the integral manifolds 𝔐(c, h) and 𝔐R(c, h). The other eight special values give rise to nine different topologically distinct cases. We give a complete description of the geometry of these sets along with their homology. These results confirm some conjectures and refutes several others.

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45

ANNABY, M. H., S. A. BUTERIN, and G. FREILING. "SAMPLING AND BIRKHOFF REGULAR PROBLEMS." Journal of the Australian Mathematical Society 87, no. 3 (December 2009): 289–310. http://dx.doi.org/10.1017/s144678870900024x.

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AbstractWe establish new sampling representations for linear integral transforms associated with arbitrary general Birkhoff regular boundary value problems. The new approach is developed in connection with the analytical properties of Green’s function, and does not require the root functions to be a basis or complete. Unlike most of the known sampling expansions associated with eigenvalue problems, the results obtained are, generally speaking, of Hermite interpolation type.

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46

Climent Vidal, J., and J. Soliveres Tur. "Birkhoff-Frink representations as functors." Mathematische Nachrichten 283, no. 5 (March 15, 2010): 686–703. http://dx.doi.org/10.1002/mana.200610832.

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47

Kehoe, Elaine. "2015 AMS-SIAM Birkhoff Prize." Notices of the American Mathematical Society 62, no. 04 (April 1, 2015): 423–24. http://dx.doi.org/10.1090/noti1233.

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48

Kehoe, Elaine. "2012 AMS--SIAM Birkhoff Prize." Notices of the American Mathematical Society 59, no. 04 (April 1, 2012): 1. http://dx.doi.org/10.1090/noti827.

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49

Chmieliński, Jacek, and Paweł Wójcik. "Approximate symmetry of Birkhoff orthogonality." Journal of Mathematical Analysis and Applications 461, no. 1 (May 2018): 625–40. http://dx.doi.org/10.1016/j.jmaa.2018.01.031.

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50

Kacsó, Daniela P., and Heinz H. Gonska. "Simultaneous Approximation by Birkhoff Interpolators." Rocky Mountain Journal of Mathematics 28, no. 4 (December 1998): 1303–20. http://dx.doi.org/10.1216/rmjm/1181071718.

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