Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen / Sophus Lie ; bearbeitet und herausgegeben von Georg Scheffers.

Lie, Sophus; Scheffers, Georg

doi:10.5962/bhl.title.18549

Cited by 151 publications

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“…[15,16] Here we briefly summarize some fundamental properties of hyperbolic numbers. This number system has been introduced by S. Lie [11] as a two dimensional example of the more general class of the commutative hypercomplex numbers systems [13]. Now let us introduce a hyperbolic plane on the analogy of the Gauss-Argand plane of the complex variable.…”

Section: Basic Definitionsmentioning

confidence: 99%

“…Even if Minkowski gave a geometrical interpretation of the special relativity space-time shortly after (1907-1908) Einstein's fundamental paper, a mathematical tool exploitable in the context of the Minkowski space-time has begun to be carried out only few decades ago ( [5]- [10]). This mathematical tool is based on the use of hyperbolic numbers, introduced by S. Lie in the late XIX century [11]. In analogy with the procedures applied in the case of complex numbers, it is possible to formalize, also for the hyperbolic numbers, a space-time geometry and a trigonometry following the same Euclidean axiomatic-deductive method [10,12].…”

Section: Introductionmentioning

confidence: 99%

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Space-Time Trigonometry and Formalization of the “Twin Paradox” for Uniform and Accelerated Motions

Boccaletti

Catoni

2006

AACA

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-The formal structure of the early Einstein's Special Relativity follows the axiomatic deductive method of Euclidean geometry. In this paper we show the deep-rooted relation between Euclidean and space-time geometries that are both linked to a two-dimensional number system: the complex and hyperbolic numbers, respectively. By studying the properties of these numbers together, pseudo-Euclidean trigonometry has been formalized with an axiomatic deductive method and this allows us to give a complete quantitative formalization of the twin paradox in a familiar "Euclidean" way for uniform motions as well as for accelerated ones.

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Section: Basic Definitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Space-Time Trigonometry and Formalization of the “Twin Paradox” for Uniform and Accelerated Motions

Boccaletti

Catoni

2006

AACA

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“…1, requires L p (h) to be in the ideal generated by h. This condition is also only sufficient and was mentioned as early as 1878 [7]. The Lie proof rule gives Lie's criterion [15,21,25] for invariance of h = 0 and characterizes smooth invariant manifolds. The rule DW is called differential weakening [24] and covers the trivial case when the evolution constraint implies the invariant candidate; in contrast to all other rules in the table, DW can work with arbitrary invariant assertions.…”

Section: Sufficient Conditions For Invariance Of Equationsmentioning

confidence: 99%

“…Furthermore, differential cuts [23] are shown to fundamentally improve the deductive power of Lie's criterion [15].…”

Section: Introductionmentioning

confidence: 99%

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Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations

2014

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In this paper we seek to provide greater automation for formal deductive verification tools working with continuous and hybrid dynamical systems. We present an efficient procedure to check invariance of conjunctions of polynomial equalities under the flow of polynomial ordinary differential equations. The procedure is based on a necessary and sufficient condition that characterizes invariant conjunctions of polynomial equalities. We contrast this approach to an alternative one which combines fast and sufficient (but not necessary) conditions using differential cuts for soundly restricting the system evolution domain.

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Symmetry algebra classification of scalar n$$ n $$th‐order ordinary differential equations

Waqas Shah,

Mahomed,

Azad

2024

Math Methods in App Sciences

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We obtain a complete classification of scalar th‐order ordinary differential equations for all subalgebras of vector fields in the real plane. While softwares like Maple can compute invariants of a given order, our results are for a general . The cases are well‐known in the literature. Further, it is known that there are three types of th‐order equations depending upon the point symmetry algebra they possess, namely, first‐order equations which admit an infinite dimensional Lie algebra of point symmetries, second‐order equations possessing the maximum 8‐point symmetries, and higher‐order, , admitting the maximum dimensional point symmetry algebra. We show that scalar th‐order equations for do not admit maximally an dimensional real Lie algebra of point symmetries. Moreover, we prove that for , equations can admit two types of dimensional real Lie algebra of point symmetries: one type resulting in nonlinear equations which are not linearizable via a point transformation and the second type yielding linearizable (via point transformation) equations. Furthermore, we present the types of maximal real dimensional and higher than ‐dimensional point symmetry algebras admissible for equations of order and their canonical forms. The types of lower‐dimensional point symmetry algebras which can be admitted are shown, and the equations are constructible as well. We state the relevant results in tabular form and in theorems.

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Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen / Sophus Lie ; bearbeitet und herausgegeben von Georg Scheffers.

Cited by 151 publications

References 0 publications

Space-Time Trigonometry and Formalization of the “Twin Paradox” for Uniform and Accelerated Motions

Space-Time Trigonometry and Formalization of the “Twin Paradox” for Uniform and Accelerated Motions

Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations

Symmetry algebra classification of scalar n$$ n $$th‐order ordinary differential equations

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