Let M be a compact Riemannian manifold whose geodesic flow φi : UM→UM on the unit tangent bundle is of Anosov type. In this paper we count the number of φi-closed orbits and study the distribution of prime closed geodesies in a given homology class in H1(M, Z). Here a prime closed geodesic means an (oriented) image of a φi-closed orbit by the projection p : UM → M.

The Enriques-Fano classification ([E.F], [F]) of the maximal connected algebraic subgroups of the three variable Cremona group, despite of its group theoretic feature, seems to be the most significant result on the rational threefolds so far known. In this paper as in [MU] we interpret the Enriques-Fano classification from a geometric view point, namely the geometry of minimal rational threefolds. We explained in [MU] the link between the two objects; the maximal algebraic subgroups and the minimal rational threefolds. Let (G, X) be a maximal algebraic subgroup of three variable Cremona group. We denote by ℓ(G, X) the set of all the algebraic operations (G, Y) such that Y is non-singular and projective and such that (G, Y) is isomorphic to (G, X) as law chunks of algebraic operation: namely (G, Y) is birationally equivalent to (G, X). Then we define an order in ℓ(G, X): for (G, Z), (G, W) ∊ ℓ(G, X), (G, Z)>(G, W) if there exists an G-equivariant birational morphism of Z onto W.

Let V be an irreducible non degenerate variety in Pn; a classical geometric result says that degree (V) ≥ codim V + 1 and, if equality holds, V is said to be of minimal degree. Varieties of minimal degree has been classified by Del Pezzo and Bertini and they all are intersections of quadrics. The local version of this result is due to J. Sally who proved that if is a regular local ring and is a Cohen-Macaulay local ring of minimal multiplicity, according to the bound e(R) ≥ height (I) + 1 given by Abhyankar, then the tangent cone of R is intersection of quadrics and it is Cohen-Macaulay.

Let N denote the set of non-negative integers. An affine semigroup is a finitely generated submonoid S of the additive monoid Nm for some positive integer m. Let k[S] denote the semigroup ring of S over a field k. Then one can identify k[S] with the subring of a polynomial ring k[t1, …, tm] generated by the monomials .

Let D be a bounded domain in the Euclidean space Rn (n ≧ 2) and L a uniformly elliptic partial differential operator of second order with α-Hölder continuous coefficients (0 < α ≦ 1) on D.

Let A be a noetherian ring and let I be an ideal of A contained in the Jacobson radical of A: Rad (A). We assume that the form ring of A with respect to the ideal I: G = Gr (A, I), is a normal integral domain. Hence A is a normal integral domain and one can ask for the links between Cl(A) and Cl(G).

Let C1[0, T] denote (one-parameter) Wiener space; that is the space of continuous functions x on [0, T] such that x(0) = 0. In a recent expository essay [21], Nelson calls attention to some functions on Wiener space which were discussed in the book of Feynman and Hibbs [13, section 3-10] and in Feynman’s original paper [12, section 13]. These functions have the form

Recently P. L. Lions has demonstrated the connection between the value function of stochastic optimal control and a viscosity solution of Hamilton-Jacobi-Bellman equation [cf. 10, 11, 12]. The purpose of this paper is to extend partially his results to stochastic differential games, where two players conflict each other. If the value function of stochatic differential game is smooth enough, then it satisfies a second order partial differential equation with max-min or min-max type nonlinearity, called Isaacs equation [cf. 5]. Since we can write a nonlinear function as min-max of appropriate affine functions, under some mild conditions, the stochastic differential game theory provides some convenient representation formulas for solutions of nonlinear partial differential equations [cf. 1, 2, 3].