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  • Cambridge University Press (CUP)  (5)
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  • Cambridge University Press (CUP)  (5)
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  • 1
    Online Resource
    Online Resource
    COMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR
    Cambridge University Press (CUP) ; 2023
    In:  The Journal of Symbolic Logic Vol. 88, No. 2 ( 2023-06), p. 540-561
    Details
    In: The Journal of Symbolic Logic, Cambridge University Press (CUP), Vol. 88, No. 2 ( 2023-06), p. 540-561
    Abstract: We introduce the computable FS-jump, an analog of the classical Friedman–Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).
    Type of Medium: Online Resource
    ISSN: 0022-4812 , 1943-5886
    URL: Article
    DOI: 10.1017/jsl.2022.45
    RVK:
    CA 1000
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 2023
    detail.hit.zdb_id: 2010607-5
    SSG: 5,1
    SSG: 17,1
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    Online Resource
  • 2
    Online Resource
    Online Resource
    Non-linear dynamics, chaos, complexity and enclaves in granitoid magmas
    Cambridge University Press (CUP) ; 1996
    In:  Earth and Environmental Science Transactions of the Royal Society of Edinburgh Vol. 87, No. 1-2 ( 1996), p. 217-223
    Details
    In: Earth and Environmental Science Transactions of the Royal Society of Edinburgh, Cambridge University Press (CUP), Vol. 87, No. 1-2 ( 1996), p. 217-223
    Abstract: Most natural systems display non-linear dynamic behaviour. This should be true for magma mingling and mixing processes, which may be chaotic. The equations that most nearly represent how a chaotic natural system behaves are insoluble, so modelling involves linearisation. The difference between the solution of the linearised and ‘true’ equation is assumed to be small because the discarded terms are assumed to be unimportant. This may be very misleading because the importance of such terms is both unknown and unknowable. Linearised equations are generally poor descriptors of nature and are incapable of either predicting or retrodicting the evolution of most natural systems. Viewed in two dimensions, the mixing of two or more visually contrasting fluids produces patterns by folding and stretching. This increases the interfacial area and reduces striation thickness. This provides visual analogues of the deterministic chaos within a dynamic magma system, in which an enclave magma is mingling and mixing with a host magma. Here, two initially adjacent enclave blobs may be driven arbitrarily and exponentially far apart, while undergoing independent (and possibly dissimilar) changes in their composition. Examples are given of the wildly different morphologies, chemical characteristics and Nd isotope systematics of microgranitoid enclaves within individual felsic magmas, and it is concluded that these contrasts represent different stages in the temporal evolution of a complex magma system driven by nonlinear dynamics. If this is true, there are major implications for the interpretation of the parts played by enclaves in the genesis and evolution of granitoid magmas.
    Type of Medium: Online Resource
    ISSN: 1755-6910 , 1755-6929
    URL: Article
    DOI: 10.1017/S0263593300006623
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1996
    detail.hit.zdb_id: 2411260-4
    detail.hit.zdb_id: 2402633-5
    SSG: 13
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    Online Resource
  • 3
    Online Resource
    Online Resource
    Polish Metric Spaces: Their Classification and Isometry Groups
    Cambridge University Press (CUP) ; 2001
    In:  Bulletin of Symbolic Logic Vol. 7, No. 3 ( 2001-09), p. 361-375
    Details
    In: Bulletin of Symbolic Logic, Cambridge University Press (CUP), Vol. 7, No. 3 ( 2001-09), p. 361-375
    Abstract: § 1. Introduction . In this communication we present some recent results on the classification of Polish metric spaces up to isometry and on the isometry groups of Polish metric spaces. A Polish metric space is a complete separable metric space ( X, d ). Our first goal is to determine the exact complexity of the classification problem of general Polish metric spaces up to isometry. This work was motivated by a paper of Vershik [1998], where he remarks (in the beginning of Section 2): “The classification of Polish spaces up to isometry is an enormous task. More precisely, this classification is not ‘smooth’ in the modern terminology.” Our Theorem 2.1 below quantifies precisely the enormity of this task. After doing this, we turn to special classes of Polish metric spaces and investigate the classification problems associated with them. Note that these classification problems are in principle no more complicated than the general one above. However, the determination of their exact complexity is not necessarily easier. The investigation of the classification problems naturally leads to some interesting results on the groups of isometries of Polish metric spaces. We shall also present these results below. The rest of this section is devoted to an introduction of some basic ideas of a theory of complexity for classification problems, which will help to put our results in perspective. Detailed expositions of this general theory can be found, e.g., in Hjorth [2000], Kechris [1999] , [2001].
    Type of Medium: Online Resource
    ISSN: 1079-8986 , 1943-5894
    URL: Article
    DOI: 10.2307/2687754
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 2001
    detail.hit.zdb_id: 1491989-8
    SSG: 5,1
    SSG: 17,1
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  • 4
    Online Resource
    Online Resource
    Classifying Borel automorphisms
    Cambridge University Press (CUP) ; 2007
    In:  Journal of Symbolic Logic Vol. 72, No. 4 ( 2007-12), p. 1081-1092
    Details
    In: Journal of Symbolic Logic, Cambridge University Press (CUP), Vol. 72, No. 4 ( 2007-12), p. 1081-1092
    Abstract: §1. Introduction . This paper considers several complexity questions regarding Borel automorphisms of a Polish space. Recall that a Borel automorphism is a bijection of the space with itself whose graph is a Borel set (equivalently, the inverse image of any Borel set is Borel). Since the inverse of a Borel automorphism is another Borel automorphism, as is the composition of two Borel automorphisms, the set of Borel automorphisms of a given Polish space forms a group under the operation of composition. We can also consider the class of automorphisms of all Polish spaces. We will be primarily concerned here with the following notion of equivalence: D efinition 1.1. Two Borel automorphisms f and g of the Polish spaces X and Y are said to be Borel isomorphic, f ≅ g , if they are conjugate, i.e. there is a Borel bijection φ: X → Y such that φ ∘ f = g ∘ φ. We restrict ourselves to automorphisms of uncountable Polish spaces, as the Borel automorphisms of a countable space are simply the permutations of the space. Since any two uncountable Polish spaces are Borel isomorphic, any Borel automorphism is Borel isomorphic to some automorphism of a fixed space. Hence, up to Borel isomorphism we can fix a Polish space and represent any Borel automorphism as an automorphism of this space. We will use the Cantor space 2 ω (with the product topology) as our representative space.
    Type of Medium: Online Resource
    ISSN: 0022-4812 , 1943-5886
    URL: Article
    DOI: 10.2178/jsl/1203350774
    RVK:
    CA 1000
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 2007
    detail.hit.zdb_id: 2010607-5
    SSG: 5,1
    SSG: 17,1
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  • 5
    Online Resource
    Online Resource
    Weakly pointed trees and partial injections
    Cambridge University Press (CUP) ; 2008
    In:  Journal of Symbolic Logic Vol. 73, No. 1 ( 2008-03), p. 363-368
    Details
    In: Journal of Symbolic Logic, Cambridge University Press (CUP), Vol. 73, No. 1 ( 2008-03), p. 363-368
    Abstract: We define the notion of a weakly pointed tree, and characterize the amount of genericity necessary to prevent a uniformly branching tree being weakly pointed. We use these ideas to show there is no topological analogue of a measure-theoretic selection theorem of Graf and Mauldin.
    Type of Medium: Online Resource
    ISSN: 0022-4812 , 1943-5886
    URL: Article
    DOI: 10.2178/jsl/1208358757
    RVK:
    CA 1000
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 2008
    detail.hit.zdb_id: 2010607-5
    SSG: 5,1
    SSG: 17,1
    Location Call Number Limitation Availability
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    Online Resource
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