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- CONTENTS 1987 - VOLUME 305 - SECTION I - N° 13
- .......... Page(s) .......... 569
- .......... Page(s) .......... 569
- Generalized linear programming techniques are extended to the mixed integer case. Implementation of branch and bound procedures leads to solving, in every node, a number of constrained column generation subproblems, according to the criterion of the dual simplex algorithm.
- .......... Page(s) .......... 573
- The paper shows that the information necessary to compute the successive powers of an ideal in a quadratic field is contained in a certain g-adic number, where g denotes the norm of the ideal. As an application we consider computing [n] P on an elliptic curve, for all n, when P is a given point on the curve.
- Let Q be a quadratic form with rational coefficients which is Q-isotropic and of signature (n, 1). Let be a congruence subgroup of SO(Q). The group acts on the n-dimensional hyperbolic space H; let be the smallest eigenvalue 0 of - on mod . We prove that cn, where c2=3/16, c3=3/4 and c=(2n-3)/4 for n4. This generalizes earlier results of Selberg and Sarnak for n=2 and n=3.
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- Let K((t)) be the field of Laurent series in one indeterminate over a field K of characteristic different from 2 and let a be a non-square in K. It is proved that the minimal number of terms in a representation of -1 as a sum of squares in the quaternion algebra is min ((m(a)/2)+1, s(K(a))), where m(a) is the order of the quadratic form <1, -a> in the Witt group WK and s(K()) is the level of the field K(). Inparticular, this number, if finite, is either of the form 2 or of the form 2+1. This answers a question of D. Lewis [2].
- In [2], [3], Jimbo defines the q-analog U,g of the universal envelopping algebra of a C-Lie Algebra g and shows that, for g=gl(N+1), any irreducible finite dimensional representation of g can be deformed in an irreducible representation of U1g. We show that, for any simple Lie algebra g over C, all irreducible finite dimensional representations of U1g are essentially obtained in this way (after possibly tensoring by a I-dimensional representation).
- .......... Page(s) .......... 591
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- We show that for any non-degenerate critical point X0 of the regular part of Green's function relative to the Laplacian on , where is a smooth and bounded domain in RN, N5, there exists a family (
) of () such that on , and concentrates around x0 as >0 goes to 0. We prove a converse to this result. - We classify the nonnegative solutions of the equation with m, p>1 which vanish for t=0 except at x=0. Three different situations occur, depending on the value of p-m.
- .......... Page(s) .......... 599
- The Scharfetter-Gummel scheme, i. e. the most popular scheme for 1-dimensional semiconductor problems, is proved to be a classical mixed method, suitably interpreted. The same interpretation being valid in several variables, the Scharfetter-Gummel method can then be naturally extended to 2-d and 3-d problems, as it has been sought for longtime in the literature (see e.g. [6], [8]).
- .......... Page(s) .......... 605
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- We consider the evolution equation: in ; on ; y=0 on , where denotes a partition of the boundary and . Its stabilizability is studied, i. e. the existence of a feed-back F(y, y') such that, for given initial data, the solution y=y(x, t) decays exponentially in the energy space when t. If satisfies very strong geometrical hypotheses several authors have proved that the system is stabilized by the feed-back F(y')=-by' with if the partition is suitably chosen. We prove in this Note the stabilizability of the system without geometrical hypotheses for the case of dimension n=2, i. e. . The proof is based on the use of the feed-back F(y')=-by' with , b0 and b(x)=0 on the interface-points x and on the construction of energy functionals well adapted to the system.
- Differential Topology
- (see Tome 307, Series I, 1987, p. 609)
- .......... Page(s) .......... 617
- .......... Page(s) .......... 617
- Several theorems for a geometrical study of global attractors are given. And then a Baire-type theorem to relate the global (and static) aspect of attractors and their dynamical aspect.
- .......... Page(s) .......... 623
- We describe, for every value of the parameters (, ), the phase portrait of the family of differential equations ,
- (<0 and >0 fixed). This family plays an important role in the resonant elimination of couples of invariant curves of plane diffeomorphisms (see [1]).
- .......... Page(s) .......... 627
- We present asymptotic equivalents for P ; here is a sequence of independent
- identically distributed random variables (i.i.d.r.v.'s), and (xn)n is a real sequence satisfying .
- .......... Page(s) .......... 631
- We provide several characterizations of measurability for multifunctions defined on an abstract measurable space and whose values are weakly locally compact and contain no line, in a Lindelöf locally convex vector space. We especially give results about the existence of a measurable selection and of a Castaing's representation. Moreover, we show that, for this kind of multifunction, measurability is preserved under finite or countable intersection.
- .......... Page(s) .......... 635
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- We consider a controlled recurrent diffusion on . We study models where either the control or the observations or both are step functions on a sample interval. We prove the convergence of the stationary probabilities of these models to the stationary probability of the diffusion when the sampling interval goes to zero.
- Numerical Analysis
- (see Tome 307, Series I, 1987, p. 599)
- COMPTES RENDUS DE L'ACADEMIE DES SCIENCES MATHEMATIQUE 1987 - Tome 305 - Série I - n° 13 Contents with Enghish abstracts I-III
- .......... Page(s) .......... 569
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- Topologie différentielle
- (voir tome 307, série I, 1987, p. 609)
- Analyse numérique
- (voir tome 307, série I, 1987, p. 599)
- Les "Comptes Rendus de l'Académie des Sciences" figurent dans "Current Contents" et "Pascal"
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