Les problèmes de l’hydrodynamique et les modèles mathématiques

2006

AACA

-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.

“…As it is known these properties allow us to use complex numbers for representing plane vectors. In the same way hyperbolic numbers, an extension of complex numbers [5,14] defined as…”

Section: Basic Definitionsmentioning

confidence: 99%

“…As a general rule we indicate the square of the segment lengths by capital letters, and by the same small letters the square root of their absolute value [5,14] …”

Section: Basic Definitionsmentioning

confidence: 99%

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

Boccaletti

2006

AACA

“…Remark 6.1. Exploiting some classical results about the boundary behaviour of the conformal mapping function in the vicinity of the corner points [23], we can prove that the estimate of Theorem 6.1 remains true for piecewise smooth boundaries dDB, consisting of sufficiently smooth arcs (curvilinear polygons). Remark 6.2.…”

Section: 1) Taking Into Account (58) We Writementioning

confidence: 86%

Untitled

1990

Quart. Appl. Math.

Abstract. The two-dimensional, deep-water, wave-body interaction problem for a single-hulled body, floating on the free surface of an ideal liquid, is considered. The body boundary may be nonsmooth and may intersect the free surface at arbitrary angles. The existence of a unique solution representable by a multipole-series expansion is proved for all but a discrete set of oscillation frequencies. The proof is based on the property of the associated multipoles to be a basis of the space Lp(-n,0), 1 < p < 2. Strict estimates of the form Dn -0(n~a) are also obtained for the coefficients of the multipole-series expansion for piecewise smooth (0 < a < 2) and smooth (a = 2) body boundaries.1. Introduction. Consider an infinitely long, horizontal cylinder floating on the free surface of an unbounded, infinitely deep, incompressible and inviscid liquid. In this paper we are concerned with the solvability (well-posedness) of the boundary-value problems arising when the floating body is forced, by an incident harmonic wave or by an external force, to perform time-harmonic oscillations of small amplitude about its position of stable hydrostatic equilibrium.The uniqueness question for the wave-floating body interaction problem has been studied by means of two, essentially different, methodologies. The first one is of geometric character and appropriately uses Green's theorem to ensure that the only possible solution of the homogeneous problem is the trivial one. Such an approach was inaugurated by John [18] in 1950 (see also Lenoir [24], who reworked the twodimensional case, and Kleinman [21]). Recently, Simon and Ursell [33] essentially generalized John's uniqueness theorem in a manner permitting them to consider floating and/or submerged bodies of quite a general shape. In general, the uniqueness results obtained via this methodology are valid for all wavelengths but have suffered, up to now, from some geometrical restrictions concerning the shape of the floating body. Having established uniqueness, the well-posedness of the problem can be deduced by using various methods, such as the integral-equation formulation [18,21,22] or the limiting absorption principle [11,24,25].

“…In more recent years the bidimensional hypercomplex systems in general form have been considered in detail [1,2,15,13]. 2 Hypercomplex numbers [14,16] are defined by the expression:…”

Section: Basic Conceptsmentioning

confidence: 99%

“…In the same way hyperbolic numbers, an extension of complex numbers [1,2] defined as {z = x + h y; h 2 = 1; x, y ∈ R; h / ∈ R}, are related to space-time geometry [3]. Indeed their square module given by 1 |z| 2 = zz ≡ x 2 − y 2 is the Lorentz invariant of two-dimensional special relativity, and their unimodular multiplicative group is the special relativity Lorentz group [3,4].…”

Section: Introductionmentioning

confidence: 99%

Two-dimensional hypercomplex numbers and related trigonometries and geometries

Cannata

et al. 2004

AACA

All the commutative hypercomplex number systems can be associated with a geometry. In two dimensions, by analogy with complex numbers, a general system of hypercomplex numbers {z = x + u y; u 2 = α + u β; x, y, α, β ∈ R; u / ∈ R} can be introduced and can be associated with plane Euclidean and pseudo-Euclidean (space-time) geometries. In this paper we show how these systems of hypercomplex numbers allow to generalise some well known theorems of the Euclidean geometry relative to the circle and to extend them to ellipses and to hyperbolas. We also demonstrate in an unusual algebraic way the Hero formula and Pytaghoras theorem, and show that these theorems hold for the generalised Euclidean and pseudo-Euclidean plane geometries.