Academic literature on the topic 'Oka manifold'
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Journal articles on the topic "Oka manifold"
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LÁRUSSON, FINNUR. "EXCISION FOR SIMPLICIAL SHEAVES ON THE STEIN SITE AND GROMOV'S OKA PRINCIPLE." International Journal of Mathematics 14, no. 02 (March 2003): 191–209. http://dx.doi.org/10.1142/s0129167x03001727.
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A complex manifold X is said to satisfy the Oka–Grauert property if the inclusion [Formula: see text] is a weak equivalence for every Stein manifold S, where the spaces of holomorphic and continuous maps from S to X are given the compact-open topology. Gromov's Oka principle states that if X has a spray, then it has the Oka–Grauert property. The purpose of this paper is to investigate the Oka–Grauert property using homotopical algebra. We embed the category of complex manifolds into the model category of simplicial sheaves on the site of Stein manifolds. Our main result is that the Oka–Grauert property is equivalent to X representing a finite homotopy sheaf on the Stein site. This expresses the Oka–Grauert property in purely holomorphic terms, without reference to continuous maps.
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TRIVEDI, SAURABH. "STRATIFIED TRANSVERSALITY OF HOLOMORPHIC MAPS." International Journal of Mathematics 24, no. 13 (December 2013): 1350106. http://dx.doi.org/10.1142/s0129167x13501061.
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We discuss genericity and stability of transversality of holomorphic maps to complex analytic stratifications. We prove that the set of maps between Stein manifolds and Oka manifolds transverse to a countable collection of submanifolds in the target is dense in the space of holomorphic maps with the weak topology. This greatly generalizes earlier results on the genericity of transverse maps by Forstnerič and by Kaliman and Zaidenberg. As an application we show that the Whitney (a)-regularity of a complex analytic stratification is necessary and sufficient for the stability of transverse holomorphic maps between a Stein manifold and an Oka manifold. This gives an analogue of a theorem in the real case due to Trotman.
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Kusakabe, Yuta. "An implicit function theorem for sprays and applications to Oka theory." International Journal of Mathematics 31, no. 09 (July 17, 2020): 2050071. http://dx.doi.org/10.1142/s0129167x20500718.
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We solve fundamental problems in Oka theory by establishing an implicit function theorem for sprays. As the first application of our implicit function theorem, we obtain an elementary proof of the fact that approximation yields interpolation. This proof and Lárusson’s elementary proof of the converse give an elementary proof of the equivalence between approximation and interpolation. The second application concerns the Oka property of a blowup. We prove that the blowup of an algebraically Oka manifold along a smooth algebraic center is Oka. In the appendix, equivariantly Oka manifolds are characterized by the equivariant version of Gromov’s condition [Formula: see text], and the equivariant localization principle is also given.
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Kusakabe, Yuta. "Dense holomorphic curves in spaces of holomorphic maps and applications to universal maps." International Journal of Mathematics 28, no. 04 (April 2017): 1750028. http://dx.doi.org/10.1142/s0129167x17500288.
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We study when there exists a dense holomorphic curve in a space of holomorphic maps from a Stein space. We first show that for any bounded convex domain [Formula: see text] and any connected complex manifold [Formula: see text], the space [Formula: see text] contains a dense holomorphic disc. Our second result states that [Formula: see text] is an Oka manifold if and only if for any Stein space [Formula: see text] there exists a dense entire curve in every path component of [Formula: see text]. In the second half of this paper, we apply the above results to the theory of universal functions. It is proved that for any bounded convex domain [Formula: see text], any fixed-point-free automorphism of [Formula: see text] and any connected complex manifold [Formula: see text], there exists a universal map [Formula: see text]. We also characterize Oka manifolds by the existence of universal maps.
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Ramos-Peon, Alexandre, and Riccardo Ugolini. "Parametric jet interpolation for Stein manifolds with the density property." International Journal of Mathematics 30, no. 08 (July 2019): 1950046. http://dx.doi.org/10.1142/s0129167x19500460.
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Given a Stein manifold with the density property, we show that under a suitable topological condition it is possible to prescribe derivatives at a finite number of points to automorphisms depending holomorphically on a Stein parameter. This is an Oka property of the manifold and is related to its holomorphic flexibility.
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STOPAR, KRIS. "APPROXIMATION OF HOLOMORPHIC MAPPINGS ON 1-CONVEX DOMAINS." International Journal of Mathematics 24, no. 14 (December 2013): 1350108. http://dx.doi.org/10.1142/s0129167x13501085.
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Let π : Z → X be a holomorphic submersion of a complex manifold Z onto a complex manifold X and D ⋐ X a 1-convex domain with strongly pseudoconvex boundary. We prove that under certain conditions there always exists a spray of π-sections over [Formula: see text] which has prescribed core, it fixes the exceptional set E of D, and is dominating on [Formula: see text]. Each section in this spray is of class [Formula: see text] and holomorphic on D. As a consequence we obtain several approximation results for π-sections. In particular, we prove that π-sections which are of class [Formula: see text] and holomorphic on D can be approximated in the [Formula: see text] topology by π-sections that are holomorphic in open neighborhoods of [Formula: see text]. Under additional assumptions on the submersion we also get approximation by global holomorphic π-sections and the Oka principle over 1-convex manifolds. We include an application to the construction of proper holomorphic maps of 1-convex domains into q-convex manifolds.
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Lutterodt, C. H. "A meromorphic extension of oka-weil approximation in a stein manifold." Complex Variables, Theory and Application: An International Journal 16, no. 2-3 (April 1991): 153–62. http://dx.doi.org/10.1080/17476939108814477.
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Tereshina, Maria, Oxana Erina, Dmitriy Sokolov, Lyudmila Efimova, and Nikolay Kasimov. "Nutrient dynamics along the Moskva River under heavy pollution and limited self-purification capacity." E3S Web of Conferences 163 (2020): 05014. http://dx.doi.org/10.1051/e3sconf/202016305014.
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An extensive study conducted during the dry summer of 2019 provided a detailed picture of the nutrient content dynamics along the Moskva River. Water sampling at 38 locations on the main river and at 17 of its tributaries revealed a manifold increase in phosphorus and nitrogen concentrations as the river crosses the Moscow metropolitan area, which can be attributed to both direct discharge of poorly treated sewage and nonpoint urban pollution. Even at the Moskva River lower reaches, where the anthropogenic pressure on the river and its tributaries is less pronounced, the inorganic nitrogen and phosphorus content remains consistently high and exceeds the environmental guidelines by up to almost 10 times. This indicates increased vulnerability of the Moskva River ecosystem during periods of low flow, which can be a major factor of eutrophication in the entire Moskva-Oka-Volga system. Comparison of our data with some archive records shows no significant improve in the nutrient pollution of the river since the 1990s, which raises further concern about the effectiveness of water quality management in Moscow urban region.
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Forstnerič, Franc. "Oka manifolds." Comptes Rendus Mathematique 347, no. 17-18 (September 2009): 1017–20. http://dx.doi.org/10.1016/j.crma.2009.07.005.
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Lárusson, Finnur, and Tuyen Trung Truong. "Approximation and interpolation of regular maps from affine varieties to algebraic manifolds." MATHEMATICA SCANDINAVICA 125, no. 2 (October 19, 2019): 199–209. http://dx.doi.org/10.7146/math.scand.a-114893.
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We consider the analogue for regular maps from affine varieties to suitable algebraic manifolds of Oka theory for holomorphic maps from Stein spaces to suitable complex manifolds. The goal is to understand when the obstructions to approximation or interpolation are purely topological. We propose a definition of an algebraic Oka property, which is stronger than the analytic Oka property. We review the known examples of algebraic manifolds satisfying the algebraic Oka property and add a new class of examples: smooth nondegenerate toric varieties. On the other hand, we show that the algebraic analogues of three of the central properties of analytic Oka theory fail for all compact manifolds and manifolds with a rational curve; in particular, for projective manifolds.
Dissertations / Theses on the topic "Oka manifold"
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Platt, Karl Florian Erich. "Das Oka-Grauert-Prinzip für Kozyklen mit Werten in Bündeln von nicht-abelschen Gruppen." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2014. http://dx.doi.org/10.18452/16876.
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Ein bedeutender Satz von L. Bungart und H. Grauert besagt, dass, für eine Gruppe G von invertierbaren Elementen einer Banachalgebra, je zwei G-wertige holomorphe Kozyklen über einer beliebigen Steinschen Mannigfaltigkeit holomorph äquivalent sind, wenn sie dort stetig äquivalent sind. Eine einfachere Form dieses Satzes wurde erstmals von K. Oka bewiesen. Aussagen dieser Art werden deshalb auch Okasche Prinzipe oder Oka-Grauert-Prinzipe genannt. Der Bungert-Grauert-Satz ist auch in dem Fall von Bedeutung, in dem die Steinsche Mannigfaltigkeit ein Gebiet in der komplexen Ebene ist. Man kann deshalb in der Literatur auch direkte Beweise für den Spezialfall finden, in dem ein G-wertiger holomorpher, stetig trivialer Kozyklus betrachtet wird. Dieser ist, nach dem oben erwähnten Satz, dann auch holomorph trivial. Ziel dieser Dissertation ist es, den Bungart-Grauert-Satz für Gebiete in der komplexen Ebene auch im allgemeinen Fall direkt zu beweisen. Dieser direkte Beweis ist wesentlich einfacher als der bisherige und muss nicht, wie bei L. Bungart und H. Grauert, auf eine Theorie von mehreren Veränderlichen zurückgreifen. Wie in den Arbeiten von L. Bungart und H. Grauert gezeigt, kann dies durch das sogenannte Verdrillen, einer Methode aus einer allgemeinen Theorie von holomorphen Kozyklen mit Werten in Bündeln von Gruppen, erzielt werden. Der größte Teil der Dissertation besteht deshalb darin, eine solche Theorie im Fall von Gebieten in der komplexen Ebene direkt aufzubauen.An important theorem of L. Bungart and H. Grauert says that for the group G of invertible elements of a banachalgebra, two holomorphic, G-valued cocycles over a Stein manifold, which are continiously equivalent, are holomorphically equivalent there. A simpler form of that theorem was first proven by K. Oka. That''s why theorems like this are known as Oka-Grauert-priciples as well. The Bungart-Grauert theorem is also significant if the Stein manifold is a domain in the complex plane. That''s why direct proofs of the special case, in which a continiously trivial, holomorphic cocycle is considered, can also be found in literature. Following the Bungart-Grauert theorem mentioned above, such a cocycle is also holomorphically trivial. The goal of this thesis is to prove the general case of the Bungart-Grauert theorem for a domain in the complex plane directly. That direct proof is much more simple than the old one. Furthermore this direct proof doesn''t have to resort to a theory of multiple variables, unlike the proof from L. Bungart and H. Grauert does. As shown in the original works, such a proof can be archieved by using the so called twisting. Twisting is a method from a theory of holomorphic cocycles with values in bundles of groups. In the main part of this thesis such a theory is build directly for domains in the complex plane.
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Trivedi, Saurabh. "Sur les stratifications réelles et analytiques complexes (a) - régulières de Whitney et Thom." Thesis, Aix-Marseille, 2013. http://www.theses.fr/2013AIXM4719.
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En 1979, Trotman a démontré que les stratifications réelles lisses qui satisfont la condition de (a)-régularité sont précisément celles pour lesquelles la transversalité aux strates des applications est une condition stable dans la topologie forte. C'était un résultat surprenant puisque la (t)-régularité semblait être plus appropriée pour la stabilité de la transversalité, une erreur qui a été faite dans plusieurs articles avant que ce résultat soit montré par Trotman. Notre premier résultat est un analogue au résultat de Trotman pour la topologie faible.Il y a une dizaine d'années Trotman a demandé si le même résultat est valable pour les stratifications analytiques complexes. Dans ce travail on démontre un analogue du résultat de Trotman dans le cas complexe, en utilisant la notion de variété de Oka introduite par Forstneric et on montre que la conjecture n'est pas vraie en général en donnant des contre-exemples.Dans sa thèse, Trotman a formulé une conjecture pour généraliser son résultat pour les stratifications (a_f)-régulières de Thom. Dans une tentative de résolution de cette conjecture on a observé que la transversalité par rapport à un feuilletage est une condition stable, cependant ce n'est pas une condition générique. Donc, en voulant imiter la preuve de Trotman on ne pourra pas obtenir cette généralisation. Néanmoins, on donne ici une preuve de cette conjecture. Ce résultat peut être résumé en disant que les (a_f)-défauts dans une stratification peuvent être détectés en perturbant les applications transverses au feuilletage induit par f. Certaines techniques pour détecter (a_f)-défauts sont aussi données vers la finTrotman in 1979 proved that real smooth stratifications which satisfy the condition of $(a)$-regularity are precisely those stratifications for which transversality to the strata of smooth mappings is a stable condition in the strong topology. This was a surprising result since $(t)$-regularity seemed to be more appropriate for stability of transversality, a mistake that was made in several articles before this result of Trotman. Our first result is an analogue of this result of Trotman for the weak topology.Trotman asked more than ten years ago whether a similar result holds for complex analytic stratifications. We will give an analogue of Trotman's result in the complex setting using Forstneriv c's notion of Oka manifolds and show that the result is not true in general by giving counterexamples.In his Ph.D. thesis Trotman conjectured a generalization of his result for Thom $(a_f)$-regular stratifications. In an attempt to prove this conjecture we noticed that while transversality to a foliation is a stable condition, it is not generic in general. Thus, mimicking the proof of the result of Trotman would not suffice to obtain this generalization. Nevertheless, we will present a proof of this conjecture in this work. This result can be summarized by saying that Thom $(a_f)$-faults in a stratification can be detected by perturbation of maps transverse to the foliation induced by $f$. Some other techniques of detecting $(a_f)$-faults are also given towards the end
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Hanysz, Alexander. "Holomorphic flexibility properties of complements and mapping spaces." Thesis, 2013. http://hdl.handle.net/2440/82397.
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The classical Oka principle in complex analysis is a heuristic, supported by theorems of Oka, Grauert and others, to the effect that certain holomorphically defined problems involving Stein manifolds have only topological obstructions to their solution. Gromov’s influential 1989 paper on the Oka principle introduced the class of so-called elliptic manifolds, and gave an Oka principle for maps from Stein manifolds into elliptic manifolds. More recently, Forstneric and Lárusson have introduced the category of Oka manifolds and Oka maps, which fit naturally into an abstract homotopy-theoretic framework; every elliptic manifold in Gromov’s sense is also an Oka manifold. Examples of Oka manifolds are complex Lie groups and their homogeneous spaces (which are also elliptic); the complement in Pn of an algebraic subvariety of codimension at least 2; Hirzebruch surfaces; and more generally any fibre bundle whose base and fibre are Oka. The Oka property can be thought of as a sort of anti-hyperbolicity; the notion of Kobayashi hyperbolicity expresses a type of holomorphic rigidity, and conversely, Oka manifolds are those that enjoy a high degree of holomorphic flexibility. Other flexibility properties enjoyed by Oka manifolds include strong dominability: for every Oka manifold X and every p ∈ X there exists a holomorphic map Cn → X which maps 0 to p and is a submersion at 0; and C-connectedness: every pair of points can be joined by an entire curve. The aim of this thesis is to provide new examples of Oka manifolds, and to shed light on the relationship between the Oka property and other types of holomorphic flexibility. Naturally occurring candidates for examples include complements of hypersurfaces in Pn, especially low-degree or non-algebraic hypersurfaces (in contrast with Kobayashi’s conjecture that the complement of a generic high-degree hypersurface should be hyperbolic), and spaces of holomorphic maps. This thesis contains three chapters. The first chapter outlines the historical development of Oka theory, gives an overview of the remaining chapters, and suggests some directions for future research. Chapter 2 is a paper entitled Oka properties of some hypersurface complements, to appear in Proceedings of the American Mathematical Society. There are two main results: a characterisation of when a complement in Pn of hyperplanes is Oka, and the result that the complement of the affine graph of a meromorphic function is Oka, subject to some restrictions. The proof of the second result involves an extension to Gromov’s technique of localisation of algebraic subellipticity. Chapter 3 is a paper entitled Holomorphic flexibility properties of the space of cubic rational maps. Define Rd to be the space of rational functions of degree d on the Riemann sphere. Geometric invariant theory can be used to explore the structure of Rd: the Möbius group acts on Rd by precomposition and postcomposition. The two-sided action on R₂ is transitive, implying that R₂ is an Oka manifold. The action on R₃ has C as its categorical quotient; Section 3.4 gives an explicit formula for the quotient map and describes its structure in some detail. Furthermore, R₃ is strongly dominable and C-connected.Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2013
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Ritter, Tyson. "Acyclic embeddings of open Riemann surfaces into elliptic manifolds." Thesis, 2010. http://hdl.handle.net/2440/69578.
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In complex geometry, the Oka principle refers to a collection of results which state that certain holomorphically defined problems involving Stein manifolds only have topological obstructions to their solution. Such results are often surprising as it is typically much more difficult to solve a problem holomorphically than continuously, given the extra constraints that holomorphic maps must satisfy. In his seminal 1989 paper on the Oka principle, Gromov introduced the concept of an elliptic complex manifold and obtained an Oka principle for holomorphic maps from Stein manifolds into elliptic manifolds. This result, together with the more recent discovery of several stronger Oka properties that hold for such maps, establishes elliptic manifolds as objects of great interest. Yet although several important collections of elliptic manifolds have been discovered, the boundaries of the class of elliptic manifolds have not yet been fully explored or understood. In this thesis we investigate the existence of proper holomorphic embeddings of open Riemann surfaces into elliptic Stein manifolds where the embedding is acyclic, meaning that it gives a homotopy equivalence between its source and target. This is the simplest case of a more general question on the existence of acyclic proper holomorphic embeddings of Stein manifolds into elliptic Stein manifolds (open Riemann surfaces are precisely the one-dimensional Stein manifolds). A positive answer to the general question would give complete information about the possible homotopy types that elliptic manifolds may have. These questions also generalise existing results on embeddings of Stein manifolds into affine space, with links to the long-standing question of whether every open Riemann surface can be properly holomorphically embedded into C². The contributions of the thesis are contained within two papers, presented as Chapters 2 and 3. In the first paper we study acyclic embeddings of Riemann surfaces with abelian (possibly trivial) fundamental group into two-dimensional elliptic Stein manifolds. By extending recent techniques of Wold and Forstneric, we prove a strong Oka principle for embeddings of so-called circular domains into the elliptic Stein manifold CxC*. Using this result we show that every Riemann surface with abelian fundamental group properly holomorphically acyclically embeds into a two-dimensional elliptic Stein manifold. In the second paper we examine acyclic embeddings of open Riemann surfaces with arbitrary fundamental group. Using an important example of Margulis, we form new examples of elliptic manifolds by taking quotients of C³ by groups of affine transformations, and use these manifolds to obtain suitable targets for acyclic embeddings of Riemann surfaces. Our main result is that every open Riemann surface acyclically embeds into an elliptic manifold.Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2010
Book chapters on the topic "Oka manifold"
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Forstnerič, Franc. "Oka Manifolds." In Stein Manifolds and Holomorphic Mappings, 207–62. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-61058-0_5.
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Forstnerič, Franc. "Oka Manifolds." In Stein Manifolds and Holomorphic Mappings, 185–240. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-22250-4_5.
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Forstnerič, Franc. "Elliptic Complex Geometry and Oka Theory." In Stein Manifolds and Holomorphic Mappings, 263–317. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-61058-0_6.
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Forstnerič, Franc. "Elliptic Complex Geometry and Oka Principle." In Stein Manifolds and Holomorphic Mappings, 241–90. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-22250-4_6.
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Forstnerič, Franc. "Surjective Holomorphic Maps onto Oka Manifolds." In Complex and Symplectic Geometry, 73–84. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-62914-8_6.
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Forstnerič, Franc. "Applications of Oka Theory and Its Methods." In Stein Manifolds and Holomorphic Mappings, 353–402. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-61058-0_8.
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Naeve, Ambjörn. "Opportunistic (L)earning in the Mobile Knowledge Society." In Refining Current Practices in Mobile and Blended Learning, 239–58. IGI Global, 2012. http://dx.doi.org/10.4018/978-1-4666-0053-9.ch016.
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This paper discusses the concept of opportunistic collaboration within the emerging mobile knowledge society. The paper illustrates how opportunistic collaboration can be applied to the areas of learning and earning by demonstrating how to transform a traditional unemployment agency into an Opportunistic Collaboration Agency based on an entrepreneurial supply-support network structured in the form of a prosumer manifold. The OCA pattern provides ways to capture the dynamics of entrepreneurial work-relations in the emerging ’work-portfolio society’, increase the transparency of the entire value chain of economic activities, and create prosumer value loops that can support multi-dimensional bartering and increase the opportunities for marginalized groups of people to create value together. The paper ends by demonstrating how to use the OCA pattern to transform a traditional educational institution into an Opportunistic Learning Activity Broker that could help to bridge the gap between formal and informal learning.
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Patel, Dimple, and Deepti Thakur. "Managing Open Access (OA) Scholarly Information Resources in a University." In Digital Libraries and Institutional Repositories, 474–99. IGI Global, 2020. http://dx.doi.org/10.4018/978-1-7998-2463-3.ch029.
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Open Access (OA) to scholarly information has now become a reality. Due to the efforts of OA supporters worldwide now even commercial publishers have started supporting open access to their content through various open access models. Many public institutions like universities and R&D Labs have realized the importance of OA in developing the society in general. As a result, these institutions have come up with OA repositories, archives and libraries. As with any such proliferation of information, OA resources have increased manifold and can easily overwhelm even an experienced user. Also different repositories may use various digital library software, which presents the problem of multifarious search interfaces and features. The solution can be found in the open community of open source software and open standards. The open source metadata harvesting software PKP-OHS and the open protocol for metadata harvesting i.e. OAI-PMH come to the rescue. This chapter discusses how PKP-OHS was implemented as a pilot study at the Central University of Himachal Pradesh (CUHP).
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Patel, Dimple, and Deepti Thakur. "Managing Open Access (OA) Scholarly Information Resources in a University." In Scholarly Communication and the Publish or Perish Pressures of Academia, 224–55. IGI Global, 2017. http://dx.doi.org/10.4018/978-1-5225-1697-2.ch011.
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Open Access (OA) to scholarly information has now become a reality. Due to the efforts of OA supporters worldwide now even commercial publishers have started supporting open access to their content through various open access models. Many public institutions like universities and R&D Labs have realized the importance of OA in developing the society in general. As a result, these institutions have come up with OA repositories, archives and libraries. As with any such proliferation of information, OA resources have increased manifold and can easily overwhelm even an experienced user. Also different repositories may use various digital library software, which presents the problem of multifarious search interfaces and features. The solution can be found in the open community of open source software and open standards. The open source metadata harvesting software PKP-OHS and the open protocol for metadata harvesting i.e. OAI-PMH come to the rescue. This chapter discusses how PKP-OHS was implemented as a pilot study at the Central University of Himachal Pradesh (CUHP).