Rozprawy doktorskie: „Schubert”

1

Müller-Kelwing, Karin. "Franz Schubert". Böhlau Verlag, 2020. https://slub.qucosa.de/id/qucosa%3A75064.

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2

Meggison, Natalie Rebecca. "Situating Schubert's Ossian settings, music, literature, and culture (Franz Schubert, Austria)". Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/MQ60383.pdf.

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3

Mizerski, Maciej. "Singularities of Schubert varieties". Thesis, McGill University, 2000. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=33427.

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Let G be a semi-simple linear algebraic group defined over an algebraically closed field and B a Borel subgroup. A Schubert variety is the closure of an orbit of the group B in the flag variety G/B. The present thesis studies the algebraic curves with a torus action in general and in the case of Schubert varieties. It also presents two proofs of Peterson's Theorem and describes the singular locus of Schubert varieties in terms of the Peterson map.

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4

Allio, Guy. "Mozart-Schubert-Beethoven : filiations". Bordeaux 2, 1989. http://www.theses.fr/1989BOR23069.

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5

Mussard, Timothy S. "Embellishing Schubert's Songs : a performance practice /". Thesis, Connect to this title online; UW restricted, 1987. http://hdl.handle.net/1773/11217.

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6

Hengelbrock, Harald. "Symmetries in quantum Schubert calculus". [S.l.] : [s.n.], 2003. http://deposit.ddb.de/cgi-bin/dokserv?idn=967581125.

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7

Schubert, Christian [Verfasser]. "Vorteilsausgleichung und Steuern / Christian Schubert". Frankfurt a.M. : Peter Lang GmbH, Internationaler Verlag der Wissenschaften, 2018. http://d-nb.info/1173656782/34.

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8

Bressler, Paul. "Schubert calculus in generalized cohomology". Thesis, Massachusetts Institute of Technology, 1989. http://hdl.handle.net/1721.1/67103.

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9

Bjurström, John. "Skriva improvisationsmusik med Franz Schubert". Thesis, Kungl. Musikhögskolan, Institutionen för jazz, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:kmh:diva-2814.

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During this project I have studied piano music by Franz Schubert (1797-1828) to thereafter use a couple of his harmonic structures as a foundation for composing music for piano trio. To write music that encourages improvisation and interaction between the band members has been a central focus while the broad goal of the project has been to develop myself as a musician and person. The project started in the fall of 2017, a senior recital was performed in Lilla Salen at KMH the 28th of February and was followed by time for reflection which can be found documented in my paper. Something I wanted to try with this project was to create according to a partially predetermined method. I wanted Schubert’s music to have an impact on my compositions and from there I decided to transcribe harmonic structures on four of his works to rework these into new compositions. I hoped that this method would bring a forward motion to the work and in retrospect I believe it did. Composing with a few guidelines contributed with a sense of comfort to the process and made me stay on track through passages of low inspiration. Reflecting on my work I feel that it became a good closure to my studies at KMH. A hybrid process showcasing different musical expressions, skills and sides to my personality with a work method that felt exciting.

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10

Brunson, Jason Cory. "Matrix Schubert varieties for the affine Grassmannian". Diss., Virginia Tech, 2014. http://hdl.handle.net/10919/25286.

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Schubert calculus has become an indispensable tool for enumerative geometry. It concerns the multiplication of Schubert classes in the cohomology of flag varieties, and is typically conducted using algebraic combinatorics by way of a polynomial ring presentation of the cohomology ring. The polynomials that represent the Schubert classes are called Schubert polynomials. An ongoing project in Schubert calculus has been to provide geometric foundations for the combinatorics. An example is the recovery by Knutson and Miller of the Schubert polynomials for finite flag varieties as the equivariant cohomology classes of matrix Schubert varieties. The present thesis is the start of a project to recover Schubert polynomials for the Borel-Moore homology of the (special linear) affine Grassmannian by an analogous process. This requires finitizing an affine Schubert variety to produce a matrix affine Schubert variety. This involves a choice of ``window'', so one must then identify a class representative that is independent of this choice. Examples lead us to conjecture that this representative is a k-Schur function. Concluding the discussion is a preliminary investigation into the combinatorics of Gröbner degenerations of matrix affine Schubert varieties, which should lead to a combinatorial proof of the conjecture.
Ph. D.

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11

Grigorenko, Don. "The theological method of Schubert Ogden". Theological Research Exchange Network (TREN), 1986. http://www.tren.com.

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12

Raab, Michael. "Franz Schubert : instrumentale Bearbeitungen eigener Lieder /". München : W. Fink, 1997. http://catalogue.bnf.fr/ark:/12148/cb36967051k.

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13

Schubert, Andreas [Verfasser] [Akademischer Betreuer], i Prof Dr Dr [Akademischer Betreuer] Gundlach. "Entwicklung eines Organkulturmodells des embryonalen Gaumens der Maus / Andreas Schubert. Betreuer: J. Schubert ; Prof. Dr. Dr. Gundlach". Halle, Saale : Universitäts- und Landesbibliothek Sachsen-Anhalt, 2009. http://d-nb.info/1024859134/34.

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14

Bodendorff, Werner. "Die kleineren Kirchenwerke Franz Schuberts /". Augsburg : B. Wissner, 1997. http://catalogue.bnf.fr/ark:/12148/cb36996011q.

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15

Schubert-Hartmann, geb Schubert Andreas [Verfasser], Gabriela [Akademischer Betreuer] Aust i Torsten [Gutachter] Schöneberg. "Klonierung und Charakterisierung des murinen CD97 Promotors / Andreas Schubert-Hartmann, geb. Schubert ; Gutachter: Torsten Schöneberg ; Betreuer: Gabriela Aust". Leipzig : Universitätsbibliothek Leipzig, 2014. http://d-nb.info/1238692133/34.

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16

Schubert, Dietmar [Verfasser]. "Die Mandatarhaftung im Römischen Recht / Dietmar Schubert". Baden-Baden : Nomos Verlagsgesellschaft mbH & Co. KG, 2014. http://d-nb.info/1107610443/34.

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17

Eastin, Julia Downer. "Klopstock und Schubert, die Vertonung der Oden". Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0019/MQ49345.pdf.

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18

Schubert, Roman [Verfasser]. "Semiclassical localization in phase space / Roman Schubert". Ulm : Universität Ulm. Fakultät für Naturwissenschaften, 2002. http://d-nb.info/1015324126/34.

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19

Schubert, Sebastian [Verfasser]. "Methodik zur Planung vielfaltsoptimierter Produktfamilien / Sebastian Schubert". Aachen : Shaker, 2013. http://d-nb.info/1050342828/34.

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20

Schubert, Andreas [Verfasser]. "Langzeitprognose der Tako-Tsubo Kardiomyopathie / Andreas Schubert". Lübeck : Zentrale Hochschulbibliothek Lübeck, 2013. http://d-nb.info/1043999906/34.

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21

Schubert, Stephan [Verfasser]. "Klinische Aspekte in der Kinderherztransplantation / Stephan Schubert". Berlin : Medizinische Fakultät Charité - Universitätsmedizin Berlin, 2012. http://d-nb.info/1028496389/34.

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22

Schubert, Michael [Verfasser]. "Circular flows on signed graphs / Michael Schubert". Paderborn : Universitätsbibliothek, 2018. http://d-nb.info/1161798684/34.

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23

Schubert, Mario [Verfasser]. "Radioaktive Implantate für medizinische Anwendungen / Mario Schubert". Aachen : Shaker, 2008. http://d-nb.info/1164342525/34.

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24

Yoo, Hwanchul. "Combinatorics in Schubert varieties and Specht modules". Thesis, Massachusetts Institute of Technology, 2011. http://hdl.handle.net/1721.1/67820.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, June 2011.
"June 2011." Cataloged from PDF version of thesis.
Includes bibliographical references (p. 57-59).
This thesis consists of two parts. Both parts are devoted to finding links between geometric/algebraic objects and combinatorial objects. In the first part of the thesis, we link Schubert varieties in the full flag variety with hyperplane arrangements. Schubert varieties are parameterized by elements of the Weyl group. For each element of the Weyl group, we construct certain hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial if and only if the Schubert variety is rationally smooth. For classical types the arrangements are (signed) graphical arrangements coning from (signed) graphs. Using this description, we also find an explicit combinatorial formula for the Poincaré polynomial in type A. The second part is about Specht modules of general diagram. For each diagram, we define a new class of polytopes and conjecture that the normalized volume of the polytope coincides with the dimension of the corresponding Specht module in many cases. We give evidences to this conjecture including the proofs for skew partition shapes and forests, as well as the normalized volume of the polytope for the toric staircase diagrams. We also define new class of toric tableaux of certain shapes, and conjecture the generating function of the tableaux is the Frobenius character of the corresponding Specht module. For a toric ribbon diagram, this is consistent with the previous conjecture. We also show that our conjecture is intimately related to Postnikov's conjecture on toric Specht modules and McNamara's conjecture of cylindric Schur positivity.
by Hwanchul Yoo.
Ph.D.

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25

Sanders, Robinson Olivia Claire. "Towards a declamatory performance in Schubert Lieder". Thesis, Edith Cowan University, Research Online, Perth, Western Australia, 2020. https://ro.ecu.edu.au/theses/2327.

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This study reconsiders declamation in Schubert lieder performance in light of emerging historical evidence. Johann Michael Vogl’s reputedly declamatory approach has arguably been captured in surviving Diabelli editions that document his rhetorically motivated alterations and ornamentations. Similarly, Gustav Anton von Seckendorff has detailed song-like spoken declamation that manipulates pitch, rhythm and accentuation. Recordings of five modern German speakers were transcribed, analysed and used to model effective declamation in the recitation of Johann Wolfgang von Goethe’s poems ‘Erster Verlust’ and ‘Geistes Gruß.’ PRAAT speech analysis software was used to analyse the recordings. The participants’ use of stress, emphasis and rubato was extrapolated and used (in combination with the historical evidence) to speculatively recreate several declamatory performances of Schubert lieder.

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26

Heitel, Pascal [Verfasser], Manfred [Akademischer Betreuer] Schubert-Zsilavecz, Dieter [Akademischer Betreuer] Steinhilber, Manfred [Gutachter] Schubert-Zsilavecz, Dieter [Gutachter] Steinhilber i Ulrike [Gutachter] Holzgrabe. "Entwicklung antientzündlicher Liganden nukleärer Rezeptoren / Pascal Heitel ; Gutachter: Manfred Schubert-Zsilavecz, Dieter Steinhilber, Ulrike Holzgrabe ; Manfred Schubert-Zsilavecz, Dieter Steinhilber". Frankfurt am Main : Universitätsbibliothek Johann Christian Senckenberg, 2019. http://d-nb.info/1197127941/34.

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27

Cho, Hye-Won Jennifer. "Performance practice issues in Franz Schubert's Fantasy in C major ("Wanderer fantasy"), D. 7960". Diss., Restricted to subscribing institutions, 2010. http://proquest.umi.com/pqdweb?did=2023755691&sid=4&Fmt=2&clientId=1564&RQT=309&VName=PQD.

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28

HAM, INA. "FRANZ SCHUBERT'S IMPROMPTUS D.899 AND D.935: AN HISTORICAL AND STYLISTIC STUDY". University of Cincinnati / OhioLINK, 2005. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1114981145.

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Schubert, Christian [Verfasser]. "Grossflächiges Sagnac Interferometer mit kalten Atomen / Christian Schubert". Hannover : Technische Informationsbibliothek und Universitätsbibliothek Hannover (TIB), 2012. http://d-nb.info/1024389138/34.

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Schubert, Sebastian [Verfasser], i Stefan W. [Akademischer Betreuer] Hell. "LineRESOLFT microscopy / Sebastian Schubert ; Betreuer: Stefan W. Hell". Heidelberg : Universitätsbibliothek Heidelberg, 2013. http://d-nb.info/1177810417/34.

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31

Schubert, Richard [Verfasser]. "On the effective properties of suspensions / Richard Schubert". Bonn : Universitäts- und Landesbibliothek Bonn, 2019. http://d-nb.info/118185590X/34.

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32

Liu, Ricky Ini. "Specht modules and Schubert varieties for general diagrams". Thesis, Massachusetts Institute of Technology, 2010. http://hdl.handle.net/1721.1/60196.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 87-88).
The algebra of symmetric functions, the representation theory of the symmetric group, and the geometry of the Grassmannian are related to each other via Schur functions, Specht modules, and Schubert varieties, all of which are indexed by partitions and their Young diagrams. We will generalize these objects to allow for not just Young diagrams but arbitrary collections of boxes or, equally well, bipartite graphs. We will then provide evidence for a conjecture that the relation between the areas described above can be extended to these general diagrams. In particular, we will prove the conjecture for forests. Along the way, we will use a novel geometric approach to show that the dimension of the Specht module of a forest is the same as the normalized volume of its matching polytope. We will also demonstrate a new Littlewood-Richardson rule and provide combinatorial, algebraic, and geometric interpretations of it.
by Ricky Ini Liu.
Ph.D.

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33

Umutabazi, Vincent. "Smooth Schubert varieties and boolean complexes of involutions". Licentiate thesis, Linköpings universitet, Algebra, geometri och diskret matematik, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-179060.

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This thesis is composed of two papers both in algebraic combinatorics and Coxeter groups. In Paper I, we concentrate on smoothness of Schubert varieties indexed by involutions from ﬁnite simply laced types. We show that if a Schubert variety indexed by an involution of a ﬁnite and simply laced Coxeter group is smooth, then that involution must be the longest element of a parabolic subgroup. Given a Coxeter system (W, S), we introduce in Paper II the boolean complex of involutions of W as an analogue of the boolean complex of W studied by Ragnarsson and Tenner. By using discrete Morse Theory, we compute the homotopy type for a large class of W, including all ﬁnite Coxeter groups. In all cases, the homotopy type is that of a wedge of spheres of dimension |S| − 1. In addition, we provide a recurrence formula for the number of spheres in the wedge.

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34

Subbotin, Filipp. "Franz Schubert. Fantasie Poutník, op.15 (C dur)". Master's thesis, Akademie múzických umění v Praze. Hudební fakulta AMU. Knihovna, 2007. http://www.nusl.cz/ntk/nusl-79092.

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In this essay I would like to show how it is possible to work with the musical text and also with Schubert´s texts. I think this method could help a pianist to find the right ways of performance Schubert´s music.

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35

KEDIM, IMAD. "Varietes de schubert, varietes toriques et treillis distributifs". Université Louis Pasteur (Strasbourg) (1971-2008), 2000. http://www.theses.fr/2000STR13074.

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Soit g un groupe algebrique semi-simple sur un corps k algebriquement clos, p un sous-groupe parabolique et w p le quotient du groupe de weyl de g par celui de p. Fixons un poids dominant , associe a p et notons l le fibre en droite g pk g/p. Dans la premiere partie de cette these, pour tout element w de w p, nous construisons une variete projective y w (reunion de varietes toriques) telle que pour tout poids dominant associe a p, il existe un fibre en droite ample sur y w dont le polynome de hilbert est egal a celui de l restreint a la variete de schubert x (w). Dans la seconde partie, nous etudions les proprietes combinatoires de l'ordre de bruhat sur w p, p classique, et, en application, nous construisons une famille plate sur spec (kt) telle que la fibre en (t u), u = 0 est isomorphe a x(w) et la fibre speciale est egale a y w.

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36

Weller, Beth Anne. "The reception of Schubert in England, 1828-1883". Thesis, King's College London (University of London), 2015. http://kclpure.kcl.ac.uk/portal/en/theses/the-reception-of-schubert-in-england-18281883(32622a6a-5ac6-4d24-8ba3-74353cc1d32e).html.

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This thesis examines Schubert’s reception in England between the year of his death in 1828 and the publication of Sir George Grove’s music dictionary in 1883, A Dictionary of Music and Musicians (A.D. 1450-1883). At the time of Schubert’s death, he was hardly known in England. Chapter I explores the introduction of his music to England and his early fame as a song composer. As songs had a firm place in a domestic setting, this section seeks to understand how the association with the home affected his image. It further investigates the effects of English nationalism on Schubert’s acceptance. Chapter II examines the performers and music societies as well as general cultural forces which fostered Schubert’s recognition as an equally gifted song and instrumental composer. Issues of performance standards in the acceptance of his instrumental pieces are also examined. Chapter III investigates the Crystal Palace and the particular roles of George Grove and August Manns in the proliferation of Schubert’s music. Their rediscoveries of numerous substantial scores and subsequent national and even world premiere performances, along with the accompanying programme notes and later reviews in periodicals, helped to shape Schubert’s image. Chapter IV considers canon formation in nineteenth-century England and Schubert’s placement in the canon. It then appraises contemporary writings on avant-garde composers including Franz Hueffer’s 1874 book, Richard Wagner and the Music of the Future: History and Aesthetics, and the influence these had on canon formation in terms of Schubert’s reception. Chapter V analyses Schubert’s image in contemporary English writings, particularly Grove’s article on Schubert in the Dictionary, and assesses the degree to which a relationship between Schubert’s works and biography was established in this period. It seeks to understand how biography impacted on his image and the way in which his works were received in England.

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37

Schubert, Julia Victoria [Verfasser]. "Beiträge zur Optimierung polymerer Referenzmaterialien / Julia Victoria Schubert". Hannover : Gottfried Wilhelm Leibniz Universität Hannover, 2020. http://d-nb.info/120838791X/34.

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38

Kfoury, Dimitry. "Calcul de Schubert affine et formules de Pieri". Electronic Thesis or Diss., Université de Lorraine, 2020. http://www.theses.fr/2020LORR0215.

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Les formules de Pieri sont des formules qui permettent de comprendre la structure d'algèbre de cohomologie de la Grassmannienne (affine) ou même celle des variété de Drapeaux. Plusieurs sont déjà établies dans quelques types et cas particuliers. Cependant ce problème reste encore ouvert pour la plupart des cas affines, en particulier pour trouver des formules de Pieri dans "H^*(\mathcal{G}r_G)" en types "B", "C" et "D".Dans cette thèse, même si on généralise quelques résultats pour un groupe de Weyl affine non-tordu général, on explore principalement les types A et C.Dans la variété de drapeaux de type A affine, on trouve une formule pour la multiplication, dans l'algèbre de cohomologie d'une variété de drapeaux, d'un élément de la base "\xi^w" par un autre (spécial) qu'on appellera ''crochet''. On montre ce résultat en utilisant la formule de Pieri donnée par Lam et al dans \cite{insertion}.En type C affine, on propose une conjecture pour une formule de Pieri en Cohomologie, en montrant qu'elle est valide en degré 1 et ''presque'' tous les cas du degré 2. On la vérifie aussi, en testant de nombreux exemples à l'aide de l'ordinateur.En Homologie, on redémontre, en utilisant une nouvelle stratégie simplifiée, la formule de Pieri en type C \cite{lam2010schubert}. Cette nouvelle approche pourrait éventuellement être utilisée dans le but d'établir des formules en types exceptionnels.Dans les variétés de drapeaux de dimension finie, on trouve aussi une majoration des coefficients de Littlewood-Richardson et on la généralise, en tout type, pour des classes particulières qu'on appellera ''petites classes de Schubert''
Pieri's formulas are a gateway to understanding the algebra structure of the (affine) Grassmannian or even that of Flag varieties. Several are already established in a few particular types and cases. However, this problem remains open for most affine cases, especially to find Pieri formulas in "H (\mathcal{G}r_G)" in types "B", "C" and "D".In this thesis, even if some results are generalized for non-twisted affine Weyl groups, we mainly explore types A and C. In the flag variete of affine type A, we find a formula for multiplying, in the cohomology algebra of a flag variety, one element of the base "\xi^w" by another (special) element that will be called ''crochet''. This result is shown using the Pieri formula given by Lam et al in \cite{insertion}. In the affine Type "C", we propose a conjecture for a Pieri formula in Cohomology, showing that it is valid in degree "1" and "almost" all cases of degree "2". It is also checked, by testing many examples using the computer.In Homology, the Pieri formula in type C \cite{lam2010schubert}, is re-demonstrated, using a new simplified strategy. This new approach could eventually be used to establish formulas of exceptional types.In the finite dimensional flag varieties, we find an upper bound for the littlewood-Richardson's coefficients and generalize it, in all types, to particular classes that will be called ''small Schubert classes''

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39

Qin, Yijing [Verfasser], Dirk [Akademischer Betreuer] Schubert i Dirk [Gutachter] Schubert. "Fabrication, Modelling and Surface Modification of Recycled and Virgin PET Melt-spun Fibres / Yijing Qin ; Gutachter: Dirk Schubert ; Betreuer: Dirk Schubert". Erlangen : Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), 2020. http://d-nb.info/1214888534/34.

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40

Madsen, Charles Arthur. "The Schubert-Liszt transcriptions : text, interpretation, and Lieder transformation /". view abstract or download file of text, 2003. http://wwwlib.umi.com/cr/uoregon/fullcit?p3080592.

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Thesis (Ph. D.)--University of Oregon, 2003.
Typescript. Includes vita and abstract. Includes bibliographical references (leaves 477-487). Also available for download via the World Wide Web; free to University of Oregon users.

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41

Lamotte-Schubert, Manuel [Verfasser], i Christoph [Akademischer Betreuer] Weidenbach. "Automatic authorization analysis / Manuel Lamotte-Schubert. Betreuer: Christoph Weidenbach". Saarbrücken : Saarländische Universitäts- und Landesbibliothek, 2015. http://d-nb.info/1077211538/34.

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42

Schubert, Eugen [Verfasser]. "Klassifikation leicht verwundbarer Verkehrsteilnehmer mit hochauflösendem Automobilradar / Eugen Schubert". Ulm : Universität Ulm, 2018. http://d-nb.info/1178527875/34.

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43

Schubert, Holger [Verfasser]. "Datenmodell und Testmethoden für die digitale Stationsleittechnik / Holger Schubert". Aachen : Shaker, 2006. http://d-nb.info/1170532608/34.

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44

Schubert, Nicole [Verfasser]. "Charakterisierung der Antikörper-vermittelten Aggregation der Granulozyten / Nicole Schubert". Greifswald : Universitätsbibliothek Greifswald, 2013. http://d-nb.info/1044594136/34.

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45

Schubert, Andreas [Verfasser]. "Unternehmensmitbestimmung in der SE & Co. KGaA / Andreas Schubert". Baden-Baden : Nomos Verlagsgesellschaft mbH & Co. KG, 2018. http://d-nb.info/1160307652/34.

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Lisboa, Viviane de Jesus. "O problema das 4 retas do calculo de Schubert". Universidade Federal da Paraíba, 2011. http://tede.biblioteca.ufpb.br:8080/handle/tede/7351.

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In this dissertation we expose the solve the four line problem in Schubert Calculus using the Plucker embedding, giving emphasis to the study of the relative position of the four given lines in P3, this allows us to obtain an explicit description of the solution's set as well as to give the precise meaning to the notion of general position. In chapter 1, we insert the notion of projective space and other related, which are the basic notions for addressing the problem that we treat. In chapter 2, we introduce the Plucker embedding, !, which allows us to identify the set of lines that meet a xed given line l0 with the intersection of the Plucker's quadric, Q, and the tangent space of Q at !(l0). We also give the description of all the linear varieties contained in the Plucker's quadric Q. Finally, in chapter 3 we demonstrate the Theorem 3.0.3 which is a key ingredient to and solutions for our problem. Moreover, we establish a relationship between the relative position of the four given lines and their solution's set. Finally, we conclude in the appendix with the Shapiro-Shapiro conjecture in the case of the four line problem in Schubert Calculus.
Neste trabalho expomos a resolução do problema das 4 retas do Cálculo de Schubert utilizando o mergulho de Plücker, com ênfase no estudo da posição relativa das 4 retas dadas em P3, o que nos permite obter uma descrição explícita do conjunto de soluções é dar sentido preciso à noção de posição geral. No capítulo 1 inserimos a noção de espaço projetivo e outras correlatas que servirão de base no estudo do problema a ser resolvido. No capítulo 2 introduzimos o Mergulho de Plücker, ω, o qual nos permite identificar o conjunto das retas que encontram uma reta fixa l0 com a interseção da quádrica de Plücker e o espaço tangente à mesma no ponto ω l0. Além disso damos a descrição das variedades lineares contidas na quádrica de Plücker. Porém, no capítulo 3 demonstramos o Teorema 3.0.3 que é a chave para resolução do nosso problema e fazemos a descrição do conjunto solução cada para posição relativa possível das 4 retas. Concluímos com um apêndice onde tratamos da conjectura de Shapiro-Shapiro no caso do problema das quatro retas do cálculo de Shubert.

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Schubert, Katrin [Verfasser]. "Der Versuch - Überlegungen zur Rechtsvergleichung und Harmonisierung. / Katrin Schubert". Berlin : Duncker & Humblot, 2011. http://d-nb.info/1238351581/34.

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Lin, Chia-Hsing. "A Performer’s Guide to Interpretive Issues in Schubert’s Late Piano Sonatas, D. 958, D. 959 and D. 960". University of Cincinnati / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1330023742.

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Kim, SeonJu. "Challenges faced by modern violists when preparing the F. Schubert Arpeggione sonata for performance /". Thesis, Connect to this title online; UW restricted, 2000. http://hdl.handle.net/1773/11422.

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Krause, Andreas. "Die Klaviersonaten Franz Schuberts : Form, Gattung, Ästhetik /". Kassel ; Basel ; London : Bärenreiter, 1992. http://catalogue.bnf.fr/ark:/12148/cb355245956.

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