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    Wyświetlanie 1-2 z 2
    Tytuł:
    Pairing of infinitesimal descending complex singularity with infinitely ascending, real domain singularity
    Autorzy:
    Czajko, Jakub
    Powiązania:
    https://bibliotekanauki.pl/articles/1030462.pdf
    Data publikacji:
    2020
    Wydawca:
    Przedsiębiorstwo Wydawnictw Naukowych Darwin / Scientific Publishing House DARWIN
    Tematy:
    Singularity
    contravariant differential
    covariant differential
    descending infinitesimal complex singularity
    infinitely ascending real singularity
    paired dual reciprocal space
    Opis:
    Pairing of infinitesimal descending singularity of the 2D domain of complex numbers with an infinitely ascending singularity deployed in the 1D domain of real numbers, where the real singularity can be equated operationally with never-ending, whether countable or not, infinity, requires the employment of a pair of mutually dual reciprocal spaces in order for each of the spaces of the twin quasigeometric structure to be truly operational. Creation of twin quasigeometric structures comprising paired dual reciprocal spaces that are really operational and truly invertible, is the necessary condition for making the notion of operationally sound infinity viable. Although acceptance of the multispatial reality paradigm seems optional, it is shown that even performing legitimate scalar differentiation (in accordance with product differentiation rule) can yield either incomplete or incorrect evaluations of compounded scalar functions. This curious fact implies inevitable need for awareness of conceptual superiority of the multispatial reality paradigm over the former, unspoken and thus unchallenged in the past, single-space reality paradigm, in order to prevent even inadvertent creation of formwise illegitimate, or just somewhat incomplete, pseudodifferentials, which can be obtained even with the use of quite legitimate operational rules of scalar differential calculus.
    Źródło:
    World Scientific News; 2020, 144; 56-69
    2392-2192
    Pojawia się w:
    World Scientific News
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    New product differentiation rule for paired scalar reciprocal functions
    Autorzy:
    Czajko, Jakub
    Powiązania:
    https://bibliotekanauki.pl/articles/1030632.pdf
    Data publikacji:
    2020
    Wydawca:
    Przedsiębiorstwo Wydawnictw Naukowych Darwin / Scientific Publishing House DARWIN
    Tematy:
    Scalar product differentiation rules
    integral kernels
    multispatial reality paradigm
    paired dual reciprocal space
    scalar product integration rules
    single-space reality paradigm
    singularity
    Opis:
    When integral kernel of an integral transform is being formed, it should be the outcome of scalar product differentiation rule if the kernel is supposed to be eventually used as an integrand in a prospective integration. Yet it has already been shown that despite ensuing from properly performed differentiation, the resulting integral kernel contains, beside the covariant differential that is suitable for integration, also a certain contravariant term, which is not appropriate for integration in the same space as the covariant differential. But the contravariant term also can be turned into proper, though multiplicatively inverse covariant differential, if placed within a space that is reciprocal to the given primary space in which the first, covariant differential, is represented naturally. This uncharacteristic conversion of the contravariant expression from the primary space into the reciprocal covariant differential in the dual reciprocal space that is paired with the given primary space, can be considered as indirect proof that pairing of mutually dual reciprocal spaces is necessary in order to properly form operationally legitimate and geometrically valid differential structures. Consequently, the pairing of an infinitesimal descending singularity of the 2D domain of complex numbers with an infinitely ascending singularity deployed in the 1D domain of real numbers requires certain dual reciprocal spatial or quasispatial structures, for the downward transition from 2D descending complex singularities to the 1D ascending “real” singularities to be meaningfully/unambiguously implemented. Furthermore, just as integration by parts formula is a counterpart of the regular product differentiation rule, a new multispatial scalar product integration rule is proposed as a counterpart to the singlespatial product differentiation rule, and introduced by analogy to the latter, “regular” product integration rule.
    Źródło:
    World Scientific News; 2020, 144; 358-371
    2392-2192
    Pojawia się w:
    World Scientific News
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-2 z 2

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