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1.
Summary LetE be a real Hausdorff topological vector space. We consider the following binary law * on ·E:(, ) * (, ) = (, k + ) for(, ), (, ) × E where is a nonnegative real number,k andl are integers.In order to find all subgroupoids of ( ·E, *) which depend faithfully on a set of parameters, we have to solve the following functional equation:f(f(y) k x +f(x) l y) =f(x)f(y) (x, y E). (1)In this paper, all solutionsf: of (1) which are in the Baire class I and have the Darboux property are obtained. We obtain also all continuous solutionsf: E of (1). The subgroupoids of (* ·E, *) which dapend faithfully and continuously on a set of parameters are then determined in different cases. We also deduce from this that the only subsemigroup ofL n 1 of the form {(F(x 2,x 3, ,x n ),x 2,x 3, ,x n ); (x 2, ,x n ) n – 1 }, where the mappingF: n – 1 * has some regularity property, is {1} × n – 1 .We may noitice that the Gob-Schinzel functional equation is a particular case of equation (1)(k = 0, l = 1, = 1). So we can say that (1) is of Gob—Schinzel type. More generally, whenE is a real algebra, we shall say that a functional equation is of Gob—Schinzel type if it is of the form:f(f(y) k x +f(x) l y) =F(x,y,f(x),f(y),f(xy)) wherek andl are integers andF is a given function in five variables. In this category of functional equations, we study here the equation:f(f(y) k x +f(x) l y) =f(xy) (x, y f: ). (4)This paper extends the results obtained by N. Brillouët and J. Dhombres in [3] and completes some results obtained by P. Urban in his Ph.D. thesis [11] (this work has not yet been published).Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth  相似文献   

2.
We shall give a further application of Hermite-Mahler polynomials to the consideration ofp-adic exponential function. An effective lower bound is obtained for max {| – | p ,P(e )| p }, where is an algebraic number satisfying || p <p –/(p–1), and 0 is ap-adic number with | | p depending on the degree of the polynomialPZ[y]. The bound obtained implies the transcendence ofe if ap-adic number satisfying 0 < || p <p –/(p–1) is algebraic or can be well approximated by algebraic numbers.This work was carried out while the author was a research fellow of the Alexander von Humboldt Foundation.  相似文献   

3.
Summary Considerf+ ff+ (1–f2)+ f=0 together with the boundary conditionsf(0)=f(0)=0,f ()=1. If=–1,>0, arbitrary there is at least one solution which satisfies 0<f<1 on (0, ). By the additional conditionf>0 on (0, ) or, alternately 0<1, the uniqueness of the solution is demonstrated.If=1,<0, arbitrary the existence of solutions for which –1<f<0 in some initial interval (0,t) and satisfying generallyf>1 is established. In both problems, bounds forf (0) and qualitative behavior of the solutions are shown.
Sommario Si consideri il problema definito dall'equazionef+ f f+ (1–f2)+ f=0 e dalle condizioni al contornof(0)=f (0)=0,f()=1. Assumendo=–1,>0, arbitrario si dimostra che esiste almeno una soluzione che soddisfa 0<f<1 nell'intervallo (0, ). Se in aggiunta si ipotizzaf>0 in (0, ), oppure 0<=1, l'unicità délia soluzione è assicurata.Successivamente si considéra il problema di valori al contorno con=1,<0, arbitrario. In questo caso esiste un'intera classe di soluzioni che soddisfano –1<f<0 in un intorno dell'origine e tali chef>1, in generale.Di detti problemi viene studiato il comportamento délle soluzioni e vengono determinate dalle maggiorazioni e minorazioni del valoref(0).
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4.
Let t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M n , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d t and, for each x M, we define a smooth real function x (t) : (1 + i (t)), where the i(t) are the eigenvalues of AA T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d2t , restricted to v at the point x -t M n.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that x ' (t) 0 for all x M and real t; then, if x t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.  相似文献   

5.
For every irrational number [0, 1) which is not of constant type we construct aC 2-diffeomorphism of the circle with rotation number which is of type III1. This diffeomorphism can be chosen arbitrarily close to the rotationR . Our methods also allow us to construct, for every Liouville number [0, 1), aC -diffeomorphism of the circle with rotation number which is of type III1.  相似文献   

6.
Summary We study the influence of the 1 of a closed manifoldM n (n3) on the foliations ofM defined by closed differential 1-forms with Morse singularities (of index 0,n). Every nonexact form is cohomologous to a weakly complete one, that is one whose leaf space is of the same type as that of a nonsingular form. Generically, a form has compact leaves or is weakly complete. If 1 M has no quotient isomorphic to *, then every nonexact form onM is weakly complete. We also say a form is complete if every path inM is homotopic to either a path transverse to or a path contained in a leaf of . Completeness of depends only on its de R ham cohomology class. The set of complete cohomology classes depends only on 1 M and is related to finitely generated normal subgroups of 1 M with quotient . If 1 M is nilpotent (or even polycyclic), every nonexact form onM is complete. On irreducible 3-manifolds, a form is complete iff it is cohomologous to a nonsingular one.  相似文献   

7.
Summary A totally umbilical pseudo-Riemannian submanifold with the parallel mean curvature vector field is said to be an extrinsic sphere. A regular curve in a pseudo-Riemannian manifold is called a circle if it is an extrinsic sphere. LetM be ann-dimensional pseudo-Riemannian submanifold of index (0n) in a pseudo-Riemannian manifold with the metricg and the second fundamental formB. The following theorems are proved. For 0 = +1 or –1, 1 = +1, –1 or 0 (2–2 0+ 12n–2–2) and a positive constantk, every circlec inM withg(c, c) = 0 andg( c c, c c) = 1 k 2 is a circle in iffM is an extrinsic sphere. For 0 = +1 or –1 (–0n–), every geodesicc inM withg(c, c) = 0 is a circle in iffM is constant isotropic and B(x,x,x) = 0 for anyx T(M). In this theorem, assume, moreover, that 1n–1 and the first normal space is definite or zero at every point. Then we can prove thatM is an extrinsic sphere. When = 0 orn, this fact does not hold in general.  相似文献   

8.
Summary A nonlinear generalizationÊ z of Euler's series transformation is compared with the (linear) Euler-Knopp transformationE z and a twoparametric methodE . It is shown how to applyE orE , to compute the valuef(zo) of a functionf from the power series at 0 iff is holomorphic in a half plane or in the cut plane. BothE andE , are superior toÊ z . A compact recursive algorithm is given for computingE andE ,.  相似文献   

9.
Summary A generalized Stokes problem is addressed in the framework of a domain decomposition method, in which the physical computational domain is partitioned into two subdomains 1 and 2.Three different situations are covered. In the former, the viscous terms are kept in both subdomains. Then we consider the case in which viscosity is dropped out everywhere in . Finally, a hybrid situation in which viscosity is dropped out only in 1 is addressed. The latter is motivated by physical applications.In all cases, correct transmission conditions across the interface between 1 and 2 are devised, and an iterative procedure involving the successive resolution of two subproblems is proposed.The numerical discretization is based upon appropriate finite elements, and stability and convergence analysis is carried out.We also prove that the iteration-by-subdomain algorithms which are associated with the various domain decomposition approaches converge with a rate independent of the finite element mesh size.This work was partially supported by CIRA S.p.A. under the contract Coupling of Euler and Navier-Stokes equations in hypersonic flowsDeceased  相似文献   

10.
Summary In this article we develop a method of deriving asymptotic formulae for the orbital counting function for the action of certain discrete groups of isometries of simply connected negatively curved manifolds. We consider the particular case of normal subgroups 0 of a co-compact group 0 for which the quotient 0/ k . Even in the special case of manifolds ofconstant negative curvature, this leads to new results. In particular, we have asymptotic estimates for some groups which arenot geometrically finite.Oblatum III-1993 & 18-VIII-1993The first author was supported by The Royal Society through a University Research Fellowship. The second author was supported by the UK SERC under grant number GR/G51930  相似文献   

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