5,570 research outputs found

    Electron loss of fast heavy projectiles in collision with neutral targets

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    The multiple electron loss of heavy projectiles in fast ion-atom collisions has been studied in the framework of the sudden perturbation approximation. Especially, a model is developed to calculate the cross sections for the loss of any number of electrons from the projectile ion, including the ionization of a single electron and up to the complete stripping of the projectile. For a given collision system, that is specified by the (type and charge state of the) projectile and target as well as the collision energy, in fact, the experimental cross sections for just three final states of the projectile are required by this model in order to predict the loss of any number, NN, of electrons for the same collision system, or for any similar system that differs only in the energy or the initial charge state of the projectile ion. The model is simple and can be utilized for both, the projectile and target ionization, and without that large computer resources are requested. Detailed computation have been carried out for the multiple electron loss of Xe18+^{18+} and U6+, 10+, 28+^{6+,\, 10+,\, 28+} projectiles in collision with neutral Ar and Ne gas targets

    Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures

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    The paper contains two lines of results: the first one is a study of symmetries and conservation laws of gl-regular Nijenhuis operators. We prove the splitting Theorem for symmetries and conservation laws of Nijenhuis operators, show that the space of symmetries of a gl-regular Nijenhuis operator forms a commutative algebra with respect to (pointwise) matrix multiplication. Moreover, all the elements of this algebra are strong symmetries of each other. We establish a natural relationship between symmetries and conservation laws of a gl-regular Nijenhuis operator and systems of the first and second companion coordinates. Moreover, we show that the space of conservation laws is naturally related to the space of symmetries in the sense that any conservation laws can be obtained from a single conservation law by multiplication with an appropriate symmetry. In particular, we provide an explicit description of all symmetries and conservation laws for gl-regular operators at algebraically generic points. The second line of results contains an application of the theoretical part to a certain system of partial differential equations of hydrodynamic type, which was previously studied by different authors, but mainly in the diagonalisable case. We show that this system is integrable in quadratures, i.e., its solutions can be found for almost all initial curves by integrating closed 1-forms and solving some systems of functional equations. The system is not diagonalisable in general, and construction and integration of such systems is an actively studied and explicitly stated problem in the literature

    Orthogonal separation of variables for spaces of constant curvature

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    We construct all orthogonal separating coordinates in constant curvature spaces of arbitrary signature. Further, we construct explicit transformation between orthogonal separating and flat or generalised flat coordinates, as well as explicit formulas for the corresponding Killing tensors and the St\"ackel matrices.Comment: 28 pages, one figure. Comments are welcom

    Nijenhuis Geometry

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    This work is the first, and main, of the series of papers in progress dedicated to Nienhuis operators, i.e., fields of endomorphisms with vanishing Nijenhuis tensor. It serves as an introduction to Nijenhuis Geometry that should be understood in much wider context than before: from local description at generic points to singularities and global analysis. The goal of the present paper is to introduce terminology, develop new important techniques (e.g., analytic functions of Nijenhuis operators, splitting theorem and linearisation), summarise and generalise basic facts (some of which are already known but we give new self-contained proofs), and more importantly, to demonstrate that the research programme proposed in the paper is realistic by proving a series of new, not at all obvious, results

    Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type

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    We connect two a priori unrelated topics, theory of geodesically equivalent metrics in differential geometry, and theory of compatible infinite dimensional Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove that a pair of geodesically equivalent metrics such that one is flat produces a pair of such brackets. We construct Casimirs for these brackets and the corresponding commuting flows. There are two ways to produce a large family of compatible Poisson structures from a pair of geodesically equivalent metrics one of which is flat. One of these families is (n+1)(n+2)/2(n+1)(n+2)/2 dimensional; we describe it completely and show that it is maximal. Another has dimension ≤n+2\le n+2 and is, in a certain sense, polynomial. We show that a nontrivial polynomial family of compatible Poisson structures of dimension n+2n+2 is unique and comes from a pair of geodesically equivalent metrics. In addition, we generalise a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.Comment: comments are welcom

    Applications of Nijenhuis Geometry V: geodesically equivalent metrics and finite-dimensional reductions of certain integrable quasilinear systems

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    We describe all metrics geodesically compatible with a gl-regular Nijenhuis operator LL. The set of such metrics is large enough so that a generic local curve γ\gamma is a geodesic for a suitable metric gg from this set. Next, we show that a certain evolutionary PDE system of hydrodynamic type constructed from LL preserves the property of γ\gamma to be a gg-geodesic. This implies that every metric gg geodesically compatible with LL gives us a finite dimensional reduction of this PDE system. We show that its restriction onto the set of gg-geodesics is naturally equivalent to the Poisson action of Rn\mathbb{R}^n on the cotangent bundle generated by the integrals coming from geodesic compatibility

    Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous Poisson structures

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    We consider multicomponent local Poisson structures of the form P3+P1\mathcal P_3 + \mathcal P_1, under the assumption that the third order term P3\mathcal P_3 is Darboux-Poisson and non-degenerate, and study the Poisson compatibility of two such structures. We give an algebraic interpretation of this problem in terms of Frobenius algebras and reduce it to classification of Frobenius pencils, i.e. of linear families of Frobenius algebras. Then, we completely describe and classify Frobenius pencils under minor genericity conditions. In particular we show that each such Frobenuis pencil is a subpencil of a certain maximal pencil. These maximal pencils are uniquely determined by some combinatorial object, a directed rooted in-forest with edges labeled by numbers λα\lambda_\alpha's and vertices labeled by natural numbers whose sum is the dimension of the manifold. These pencils are naturally related to certain (polynomial, in the most nondegenerate case) pencils of Nijenhuis operators. We show that common Frobenius coordinate systems admit an elegant invariant description in terms of the Nijenhuis pencil.Comment: In Version v2, Theorem 4 and its proof are improved, and a mistake in Theorem 5 is corrected. In Version v3 we changed the signs in certain formulas for cosmetical reasons, to avoid multiple use of (−1)n(-1)^n, and to make the paper better compatible with arXiv:2212.01605, and also updated the reference
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