Let us consider a stochastically continuous, separable and measurable stationary Gaussian process X = {X(t), − ∞ < t < ∞} with mean zero and with the covariance function p(t) = EX(t + s)X(s). The conditions for continuity of paths have been studied by many authors from various viewpoints. For example, Dudley [3] studied from the viewpoint of ε-entropy and Kahane [5] showed the necessary and sufficient condition in some special case, using the rather neat method of Fourier series.

Let X ⊆ Pr be a reduced, irreducible and non-degenerate projective variety over an algebraically closed field K of characteristic 0. Let reg(x) be the Castelnuovo-Mumford regularity of the sheaf of ideals associated to X.

Then it is an open problem—due to D. Eisenbud (see e.g. [E-Go])—whether

(0.1) reg(X) ≤ deg(x) - codim (x) + 1,

where deg(x) denotes the degree of X and codim(x) denotes the codimension of X. In many cases, this inequality has been proven to hold true.

The general solution of an algebraic differential equation depends on the initial conditions, though it is in general too difficult to make explicit the shape of the relationship. Painlevé studied in [8] algebraic differentia] equations of second order with the general solutions depending rationally on the initial conditions and the solvability of such equations. Giving the precise definition of the notion “rational dependence on the initial conditions”, Umemura [10] revived and generalized rigorously the discussion of Painlevé in the language of modern algebraic geometry. The theorem of Umemura is as follows; Let K be a differential field extension of complex number field C generated by a finite number of meromorphic functions on some domain in C. Let y be the general solution of a given algebraic differential equation over K. Suppose that y depends rationally on the initial conditions. Then it is contained in the terminal Km of a finite chain of differential field extensions: K = K0 ⊂ K1 ⊂… ⊂Km such that each Ki is strongly normal over Ki−1.

Making use of the FBI (Fourier-Bros-Iagolnitzer) transforms we simplify the quasianalytic singular spectrum for the Fourier hyperfunctions, which was defined for distributions by Hörmander as follows; for any Fourier hyperfunction u, (x0, ξ0) does not belong to the quasianalytic singular spectrum W FM(u) if and only if there exist positive constants C, γ and N, and a neighborhood of x0 and a conic neighborhood Г of ξ0 such that

for all x ∈ U, |ξ| ∈ Γ and |ξ| ≥ N, where M(t) is the associated function of the defining sequence Mp. This result simplifies Hörmander’s definition and unify the singular spectra for the C∞ class, the analytic class and the Denjoy-Carleman class, both quasianalytic and nonquasianalytic.

In this note we consider a finite group G which satisfies the following conditions:

• (0. 1) G is a doubly transitive permutation group on a set Ω of m + 1 letters, where m is an odd integer ≥ 3,

• (0. 2) if H is a subgroup of G and contains all the elements of G which fix two different letters α, β, then H contains unique permutation h0 ≠ 1 which fixes at least three letters,

• (0. 3) every involution of G fixes at least three letters,

• (0. 4) G is not isomorphic to one of the groups of Ree type.

For a real valued periodic smooth function u on R, n ≥ 0, one defines the osculating polynomial φs (of order 2n + 1) at a point s ∈ R to be the unique trigonometric polynomial of degree n, whose value and first 2n derivatives at s coincide with those of u at s. We will say that a point s is a clean maximal flex (resp. clean minimal flex) of the function u on S1 if and only if φs ≥ u (resp. φs ≤ u) and the preimage (φ - u)-1(0) is connected. We prove that any smooth periodic function u has at least n + 1 clean maximal flexes of order 2n + 1 and at least n + 1 clean minimal flexes of order 2n + 1. The assertion is clearly reminiscent of Morse theory and generalizes the classical four vertex theorem for convex plane curves.

By a theorem of Wahl, for canonically embedded curves which are hyperplane sections of K3 surfaces, the first Gaussian map is not surjective. In this paper we prove that if C is a general hyperplane section of high genus (> 280) of a general polarized K3 surface, then the second Gaussian map of C is surjective. The resulting bound for the genus g of a general curve with surjective second Gaussian map is decreased to g > 152.

Duvaut and Lions [2] studied the field of velocities and of temperatures in a moving incompressible Bingham fluid endowed with viscosity μ(θ) depending on the temperature θ and established the existence of a weak solution in the case of a two dimensional fluid. However, the problem of uniqueness remained unsolved. The purpose of the present paper is to give an affirmative answer to the problem, that is, to show the local existence (resp. the global existence) in the time and the uniqueness of (strong) solutions in three dimensions under the conditions that (i) the time (resp. the initial velocity and the external force) and (ii) the rate of variation of the viscosity and the yield limit with respect to the temperature are both sufficiently small. It will be easily seen that the global existence and the uniqueness also hold in the two dimensional case whenever the rate (ii) is sufficiently small.

A vertex of a planar curve γ of class is a point which attains a local maximum or minimum of its curvature function. By the definition, the number of vertices are even whenever it is finite. As a generalization of famous four vertex theorem, Pinkall [P] showed that a closed curve γ has at least 4 vertices if it bounds an immersed surface, and he conjectured that γ has at least 4g + 2 vertices when the surface has genus g.

Let b0 be a positive real number and

be a Jacobi matrix. We can associate with them a Jacobi continued fraction, which will be abbreviated to a J fraction from the next section, as follows

where An(z)/Bn(z) is the n-th Padé approximant of φ(z).

In the article I: [8], we have proved the hypoellipticity of a degenerate parabolic equation of the form:

where the coefficients a(x, t), b(x,t) and c(x, t) are complex valued smooth functions. The fundamental assumption on the coefficients is that Re a(x, t) satisfies the condition of Nirenberg and Treves ([8], (1.5)). To prove the hypoellipticity we have constructed recurcively the parametrices as pseudodifferential operators with parameter. This method may be viewed as an improvement of that of [9] and [7]. We have analyzed the properties of these parametrices by estimating the symbols with parameter associated with the given operator. We shall summerize these results in §3.

In this paper we are concerned with zero-sum two-player finite horizon games for stochastic partial differential equations (SPDE in short). The main aim is to formulate the principle of dynamic programming for the upper (or lower) value function and investigate the relationship between upper (or lower) value function and viscocity solution of min-max (or max-min) equation on Hilbert space.

Our purpose in this note is to establish a classification theorem for fiberings with a loop-space as fibre. This is deduced by applying a generalized suspension theorem which will be proved in § 1. As a by-product we obtain a proposition concerning fiberings with a loop-space as the total. Throughout this note we shall denote by the category of spaces having the based homotopy type of a CW-complex.

Dans une Note parue dans ce Journal, M. Matsushima [1] a démontré le

Théorème. Dans un espace compact de Kähler-Einstein dont la courbure est positive, un vecteur analytique contrevariant u peut s’écrire d’une et d’une seule manière sous la forme u = v + Fw où v et w sont tous les deux les vecteurs de Killing et F le tenseur définissant la structure complexe de l’espace.

In this article we want to analyze the structure of a 4 dimensional projective variety X which has a smooth ample divisor A that is a P1 bundle π : A→S over a smooth surface S.

In [Fa+So], as a consequence of a more general result, the first and third authors determined the structure of X in the case the base S of the P1 bundle A has a cover with h2,0()≠0. Here we look at the remaining cases except for those surfaces which are the projectivization of a stable rank two vector bundle over a curve (the result is obviously true for S rational).

In this paper, we give explicit combinatorial descriptions for toric extremal contractions under the relative setting, where varieties are not complete. It is well-known that the complete case is settled by using Reid’s wall theory which can not be applied to the non-complete case. Therefore, we can achieve them by using the notion of extremal primitive relations. As applications, we can generalize some of Mustaţă’s results related to Fujita’s conjecture on toric varieties for the relative case.

Let K be a compact set in a complex metrizable locally convex space E and F a complex Banach space. The study of the space of holomorphic germs ℋ(K; F) endowed with the Nachbin topology was undertaken by several authors.

Let K be a Galois extension of an algebraic number field k of finite degree with Galois group g. Then g acts on a congruent ideal class group of K as a group of automorphisms, when the class field M over K corresponding to is normal over K. Let Ig be the augmentation ideal of the group ring Zg over the ring of integers Z, namely Ig be the ideal of Zg generated by σ − 1, σ running over all elements of g. Then is the group of all elements aσ-1 where a and σ belong to and g respectively.

The 2g theta constants of second kind of genus g generate a graded ring of dimension g(g + 1)/2. In the case g ≥ 3 there must exist algebraic relations. In genus g = 3 it is known that there is one defining relation. In this paper we give a relation in the case g = 4. It is of degree 24 and has the remarkable property that it is invariant under the full Siegel modular group and whose Φ-image is not zero. Our relation is obtained as a linear combination of code polynomials of the 9 self-dual doubly-even codes of length 24.

Let H be a simplicial semigroup. We consider the semigroup ring k[H] and its defining ideal IH. For definition see the first paragraph of Section 1.

When dim(k[H]) = 1, the defining ideal IH of k[H] has been studied by many authors (e.g. [1], [2], [8], [11], [3]). In this paper, we study the ideal IH using the notion of Gröbner bases for arbitrary dimension.