G. Birkhoff: Lattice theory, 2nd ed., New York, 1948.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the present section we de ne a class of completely integrable billiard tables of dimension 2. The construction we propose is in uenced from the classi cation theorems of Liouville surfaces given in [9, 10, 12, 3, 14] and the classical examples of integrable billiards described in Section 3 (see [1, 16, 2, 5, 18, 4, 17]) By de nition, a Liouville surface is a complete 2 dimensional Riemannian manifold without boundary the geodesic ow of which admits a quadratic in velocities integral functionally independent of the energy integral. The idea to use special covers rst appears in [19, 12] see Section 3 and [3] ....

....of X, 10 where H stands for the Hamiltonian function on T X corresponding to the Riemannian metric g via the Legendre transformation. Set = f(x; 2 S Xj : h ; n(x)i 0g, n(x) being the unit inward normal to at x. The corresponding billiard ball map B is de ned as follows ([1]) Take (x; in the interior and denote by 2 x the co vector uniquely determined by j Tx = Consider the integral curve exp(tX H ) x; of the Hamiltonian vector eld X H starting at (x; If it intersects transversally S Xj at a time t 1 0 and lies entirely ....

G. Birkhoff: Dynamical systems, Amer. Math. Soc. Colloq. Publ. 9, AMS, New York, 1927

....matrices P from SL(n; Z) with such a property is isomorphic to M n Gammar Thetar (Z) Phi SL(n Gamma r; Z) where M n Gammar Thetar (Z) stands for the group of n Gamma r Theta r matrices with integral entries. In order to prove the theorem we recall the following well known assertion (cf. [30] for more general statements on sequences of maps for finitely generated abelian groups) PERTURBATIONS OF VECTOR FIELDS ON TORI 5 Lemma 2.2. Let A ae B : Z n be a subgroup. Set C = A Z = f 2 Z n : Ag. Then B = A Phi C (2.4) holds if and only if the sequence 0 A id B ....

....for the orthogonal projection on C (respectively, the identity map) Proof. The existence of a basis follows from the well known fact that any subgroup A (in our case A = Gamma Z ) of a finitely generated abelian group B (in our case B = Z n ) is also a finitely generated abelian group cf. [30]. Define j : R n R n by j ( e j e j = j e j , where e j is the j th unit vector. Without loss of generality we may assume that for each j 2 f1; ng we have j k 6= 0 for some k, otherwise we are reduced to a space of dimension n Gamma 1. Because Gamma Z ....

S. MacLane and G. Birkhoff, Algebra, MacMillan, New York 1979. MR 80d:00002

....Conclusions are drawn in Section 8. 2 Preliminaries Let X and Y be sets and f : X Gamma Y be a function. Let P(X) denote the power set of the set X . Let f Gamma1 : P(Y ) Gamma P(X) Y 0 7 Gamma f Gamma1 (Y 0 ) fx 2 X j f(x) 2 Y 0 g be the inverse image map of f . Following [MB88], let f : P(X) Gamma P(Y ) X 0 7 Gamma f (X 0 ) ff(x) 2 Y j x 2 X 0 g be the induced map of f on power sets. Thus (f ) Gamma1 and (f Gamma1 ) are both maps from PP(Y ) to PP(X) In general these two functions are different. If f is surjective then, for all Z 2 PP(Y ) f ....

S. MacLane and G. Birkhoff. Algebra. Chelsea, New York, 1988.

....resolved. Moreover, it is not easy to see if they could be solved even within some possibly extended framework. 3.1 Basics Before presenting our arguments against the formalization style of object algebras, we would like to establish some basic notions. Precise definitions can be found, e.g. in [Cohn81, MaBi79]) An algebra is a pair E;O consisting of a set E of elements (which are neither functions nor operators) and a set of operators O. Each operator o 2 O is a (partial) function mapping element(s) of E into an element of E. The set of operators may be infinite, and the set E may be subdivided ....

S. MacLane, G. Birkhoff. Algebra. MacMillan, New York, 1979

....in this section we address the problems explicitly and point out that they are not solved. Some of them could not be solved even within possibly extended framework. 3.1 Basics Before presenting our arguments we would like to establish basic notions. Precise definitions can be found, e.g. in [Cohn81, MaBi79]) An algebra is a pair E;O consisting of a set E of elements (which are neither functions nor operators) and a set O of operators over E. Each operator o 2 O is a (partial) function mapping element(s) of E into an element of E. The set of operators may be infinite, and the set E may be ....

S. MacLane, G. Birkhoff. Algebra. MacMillan, New York, 1979.

....kinds of conjugacy operators. 4. Calculus The connection between an antihomogeneous conjugacy operator and multiplication by a positive number is given in the definition of an antihomogeneous conjugacy operator. It is easy to verify using the simplest properties of lattices (see for example [3]) that the following is true. Proposition 4.1. Let P : U V be an antihomogeneous conjugacy operator and A a subset of a c 2 lattice U . Then P (sup u2A u) inf u2A P (u) P ( inf u2A u) sup u2A P (u) Summation is not present in the definitions both of a c 2 lattice and an ....

G. Birkhoff: Lattice theory, 2nd ed., New York, 1948.

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G. Birkhoff: Lattice theory, 2nd ed., New York, 1948.

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