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    Wyświetlanie 1-8 z 8
    Tytuł:
    Finegrained 3D differential operators hint at the inevitability of their dual reciprocal portrayals
    Autorzy:
    Czajko, Jakub
    Powiązania:
    https://bibliotekanauki.pl/articles/1065412.pdf
    Data publikacji:
    2019
    Wydawca:
    Przedsiębiorstwo Wydawnictw Naukowych Darwin / Scientific Publishing House DARWIN
    Tematy:
    Finegrained differential operators
    paired dual reciprocal spaces
    Opis:
    Extending differential operators during transition from 3D operations to prospective 4D operations imply the need to expand their 4D range far beyond the usual set-theoretical universe from which subsets of the domain are composed. The prospective infrastructural expansion related to the attempted extending of operations virtually requires an extra dual reciprocal space and thus implies presence of a certain multispatial structure of both the mathematical- and the corresponding to it physical reality. At this point the implication is only operational for it is deduced from the attempted extension of operational domain/scope of geometric differential operator, which, in turn, demands an expansion of their range. The necessary presence of an extra space is not being postulated but emerges from comparative evaluations of differential operators. The operational necessity of presence of paired dual reciprocal spaces or quasispatial structures also generalizes contravariance for multispatiality.
    Źródło:
    World Scientific News; 2019, 132; 98-120
    2392-2192
    Pojawia się w:
    World Scientific News
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Multiplicative inversions involving real zero and neverending ascending infinity in the multispatial framework of paired dual reciprocal spaces
    Autorzy:
    Czajko, Jakub
    Powiązania:
    https://bibliotekanauki.pl/articles/1031265.pdf
    Data publikacji:
    2021
    Wydawca:
    Przedsiębiorstwo Wydawnictw Naukowych Darwin / Scientific Publishing House DARWIN
    Tematy:
    Multiplicative inversions
    dual reciprocal spaces
    multispatial algebraic structures
    Opis:
    Inverses of complex numbers and of analytic functions are composites of mixed type for they are multiplicative inverses (i.e. reciprocals) of the modulus/magnitude combined with additive reverses of the argument/angle. Hence, the mixed inverses in the complex domain ℂ are not really reciprocals and therefore their lack of truly multiplicative reciprocity was a contributing reason that spurred the – unnecessary though still ongoing – prohibition of division by zero which is the natural reciprocal of the neverending ascending real infinity. Truly reciprocal algebraic operations are presented (via multiplicative algebraic inversions) by few examples within the new multispatial framework in terms of their abstract algebraic representations subscripted by the native algebraic bases of the mutually paired dual reciprocal (even though algebraic) spaces in which the inversive operations are performed.
    Źródło:
    World Scientific News; 2021, 151; 1-15
    2392-2192
    Pojawia się w:
    World Scientific News
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Multiplicative counterpart of the essentially additive Borsuk-Ulam theorem as the pivoting gateway to equidimensional paired dual reciprocal spaces
    Autorzy:
    Czajko, Jakub
    Powiązania:
    https://bibliotekanauki.pl/articles/1031786.pdf
    Data publikacji:
    2020
    Wydawca:
    Przedsiębiorstwo Wydawnictw Naukowych Darwin / Scientific Publishing House DARWIN
    Tematy:
    Borsuk-Ulam theorem
    dual multispatial structures
    multispatial mathematical reality
    Opis:
    Multiplicative counterpart of the essentially additive Borsuk-Ulam theorem is proposed as abstract guiding principle suitable for explanation of intricate mathematical – that is both operational/algebraic and structural/geometric – relationships existing between spaces of equal dimensionality forming dual multispatial structures, which comprise twin equidimensional dual reciprocal spaces. Under auspices of the multispatial reality paradigm, the multiplicatively inversive type of the Borsuk-Ulam theorem can thus be also interpreted as an interspatial pivoting gateway between paired dual reciprocal spaces.
    Źródło:
    World Scientific News; 2020, 150; 118-131
    2392-2192
    Pojawia się w:
    World Scientific News
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Dual Reciprocal Scalar Potentials Paired via Differential Operators in Frenet Frames Make the Operators to Act Simultaneously in Each of Two Paired 3D Reciprocal Spaces
    Autorzy:
    Czajko, Jakub
    Powiązania:
    https://bibliotekanauki.pl/articles/1046557.pdf
    Data publikacji:
    2019
    Wydawca:
    Przedsiębiorstwo Wydawnictw Naukowych Darwin / Scientific Publishing House DARWIN
    Tematy:
    3D intraspatial and interspatial duality principles
    3D potentials in inertially moving trihedron
    Paired 3D dual reciprocal spaces
    scalar covariant differential operator SCovar
    Opis:
    Extending the operating domain of 3D geometric differential operator expands also the range of its operations onto paired 3D dual reciprocal spaces as well as the scope of their validity into paired 3D multispatial structures. Intraspatial duality principle for paired 3D dual reciprocal spaces is inferred from differential operations performed on the dual reciprocal 3D spaces. The new scalar differential operator SCovar as multiplicative inverse of the scalar gradient differential operator SGrad is proposed here to deliver scalar components of covariant differentials in order to accommodate both the operational and structural legitimacy of differential and integral operations performed in 3D dual reciprocal spaces. From preliminarily formulated abstract intraspatial duality principle a generalized interspatial duality principle is deduced and the connection of paired multispatial structures established. It is shown that the finegrained geometric differential operator GDiff acts simultaneously in each of the paired dual reciprocal spaces, which is its formerly unknown operational feature.
    Źródło:
    World Scientific News; 2019, 137; 96-118
    2392-2192
    Pojawia się w:
    World Scientific News
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Unrestricted division by zero as multiplication by the – reciprocal to zero – infinity
    Autorzy:
    Czajko, Jakub
    Powiązania:
    https://bibliotekanauki.pl/articles/1030823.pdf
    Data publikacji:
    2020
    Wydawca:
    Przedsiębiorstwo Wydawnictw Naukowych Darwin / Scientific Publishing House DARWIN
    Tematy:
    Division by zero
    multiplication by infinity
    paired dual reciprocal spaces
    Opis:
    It is demonstrated that quite unrestricted operation of conventional division by the real number zero can be implemented via multiplication by the – reciprocal to zero – ascending infinity in paired dual reciprocal spaces, provided that the “real” zero and the infinity are mutually reciprocal (i.e. multiplicatively inverse). Since the infinity is not an absolute concept independent of the particular circumstances in which it is being determined, the value of the setvalued infinity is not fixed but depends on an influence function that is usually applied for evaluation of integral kernels, the operations proposed here are always defined relative to a certain abstract influence function. The conceptual and operational validity of the proposed unrestricted division and multiplication by zero, is authenticated operationally by fairly simple example, using an evaluation of Frullani’s integral.
    Źródło:
    World Scientific News; 2020, 145; 180-197
    2392-2192
    Pojawia się w:
    World Scientific News
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    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
    Tytuł:
    Wave-particle duality of a 6D wavicle in two paired 4D dual reciprocal quasispaces of a heterogeneous 8D quasispatial structure
    Autorzy:
    Czajko, Jakub
    Powiązania:
    https://bibliotekanauki.pl/articles/1062921.pdf
    Data publikacji:
    2019
    Wydawca:
    Przedsiębiorstwo Wydawnictw Naukowych Darwin / Scientific Publishing House DARWIN
    Tematy:
    Wave-particle duality
    multispatial mathematical reality of physics
    wavicle
    Opis:
    According to Louis de Broglie, wave-particle duality can imply presence of a single entity also known as wavicle. Hence, instead of the double solution he proposed to explain the duality, I depict the 6D wavicle in a multispatial mathematical entity represented in two 4D quasigeometric heterogeneous structures comprising two paired 3D dual reciprocal spaces, one of which represents the particle and the other the wave that was supposed to guide the particle in his pilot wave theory. The operational nature of the (2⨯4)D = 8D biquaternionic quasigeometric structure of the wavicle is synthesized from twin operations performed over paired 3D homogeneous dual reciprocal spaces, each of which is immersed in a pair of two 4D asymmetrically overlapping heterogeneous quasigeometric (3+1)D = 4D spatial structures.
    Źródło:
    World Scientific News; 2019, 127, 1; 1-55
    2392-2192
    Pojawia się w:
    World Scientific News
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-8 z 8

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