This paper contains a new proof of the classification of prime order elements of Bir(ℙ2) up to conjugation. The first results on this topic can be traced back to classic works by Bertini and Kantor, among others. The innovation introduced by this paper consists of explicit geometric constructions of these Cremona transformations and the parameterization of their conjugacy classes. The methods employed here are inspired to [4], and rely on the reduction of the problem to classifying prime order automorphisms of rational surfaces. This classification is completed by combining equivariant Mori theory to the analysis of the action on anticanonical rings, which leads to characterize the cases that occur by explicit equations (see [28] for a different approach). Analogous constructions in higher dimensions are also discussed.

In the study of moduli of polarized abelian varieties and of curves as well as in the theory of complex multiplications, the notion of fields of moduli for structures plays an essential role. This notion was first introduced by Matsusaka [7] for polarized varieties with some pleasing properties and later was given a more comprehensible treatment by Shimura [10] in the case of polarized abelian varieties or polarized abelian varieties with some further structures. Both authors discussed fields of moduli not only in algebraic geometry of characteristic zero but also in that of positive characteristic, but in the latter case the definition of fields of moduli seems somewhat artificial and there have been no essential applications of them so far.

Every Riemannian manifold has a unique affine connection without torsion, which is necessarily invariant by any isometrical transformation of the manifold. However, an affine transformation (i.e., transformation leaving invariant the affine connection) is not necessarily an isometrical transformation. (Consider, for example, the ordinary Euclidean space).

In [2], chap. IV, 2me partie, (7.4.8), Grothendieck considered the following problem: is any m-adic completion of an excellent ring A also excellent?

In [8] I proved that, if A is an algebra of finite type over an arbitrary field k, the answer is positive.

In this paper we consider degenerate parabolic equations, and obtain an interior and a boundary Harnack inequalities for nonnegative solutions to the degenerate parabolic equations. Furthermore we obtain boundedness and continuity of the solutions.

Let (A, m, k) be a Noetherian local ring and I an ideal of A. We set and call this graded A-algebra the Rees algebra of I. The importance of the Rees algebra R(I) is in the fact that Proj R(I) is the blowing up of Spec A with center in V(I). The Cohen-Macaulayness of Rees algebras was studied by many mathematicians. In [GS] S. Goto any Y. Shimoda gave a criterion for R(m) to be Cohen-Macaulay under the assumption that A is Cohen-Macaulay. Their results have been generalized to R(I) in [HI].

This paper has been motivated by various problems and results in differential geometry. The main motivation is the study of curvature homogeneous Riemannian spaces initiated in 1960 by I.M. Singer (see Section 9—Appendix for the precise definitions and references). Up to recently, only sporadic classes of examples have been known of curvature homogeneous spaces which are not locally homogeneous. For instance, isoparametric hypersurfaces in space forms give nice examples of nontrivial curvature homogeneous spaces (see [FKM]). To study the topography of curvature homogeneous spaces more systematically, it is natural to start with the dimension n = 3. The following results and problems have been particularly inspiring.

Let A be a Noetherian local ring of dimension d and with maximal ideal m. Then A is called Buchsbaum if every system of parameters is a weak sequence. This is equivalent to the condition that, for every parameter ideal q, the difference is an invariant I(A) of A not depending on the choice of q. (See Section 2 for the detail.) The concept of Buchsbaum rings was introduced by Stückrad and Vogel [8], and the theory of Buchsbaum singularities is now developing (cf. [6], [7], [9], [10], and [12]).

For any set S and any entire function f let

where each zero of f — a with multiplicity m is repeated m times in Ef(S) (cf. [1]). It is assumed that the reader is familiar with the notations of the Nevanlinna Theory (see, for example, [2]). It will be convenient to let E denote any set of finite linear measure on 0 < r < ∞, not necessarily the same at each occurrence. We denote by S(r, f) any quantity satisfying .

In this paper, we study some spectral properties of the discrete Schrödinger operator = Δ + q defined on a locally finite connected graph with an automorphism group whose orbit space is a finite graph.

The discrete Laplacian and its generalization have been explored from many different viewpoints (for instance, see [2] [4]). Our paper discusses the discrete analogue of the results on the bottom of the spectrum established by T. Kobayashi, K. Ono and T. Sunada [3] in the Riemannian-manifold-setting.

P. Lévy introduced, in his celebrated books [21] and [22], an infinite dimensional Laplacian called the Lévy Laplacian in connection with a number of interesting topics in variational calculus. One of the most significant features of the Lévy Laplacian is observed when it acts on the singular part of the second functional derivatives. For this reason the Lévy Laplacian has become important also in white noise analysis initiated by T. Hida [12]. On the other hand, as was pointed out by H. Yoshizawa [29], infinite dimensional rotation groups are profoundly concerned with the structure of white noise, and therefore, play essential roles in certain problems of stochastic calculus. Motivated by these works, we aim at developing harmonic analysis on infinite dimensional spaces by means of the Lévy Laplacian and infinite dimensional rotation groups.

Here we shall prove that Painlevé’s first transcendent, a solution of the equation y″ = 6y2 + x, can not be described as any combination of solutions of first order algebraic differential equations and those of linear differential equations. This result gives an answer to the question whether the function is truely new or not.

The first author and S. Takenaka introduced the structure of a Gel’fand triplet ℋ ⊂ (L2) ⊂ ℋ* into Hida’s calculus on generalized Brownian functionals [4-7]. They showed that the space ℋ of testing random variables has nice properties. For example, ℋ is closed under multiplication of two elements in ℋ, each element of ℋ is a continuous functional on the basic space ℰ*, in addition it can be considered as an analytic functional, and moreover exp [tΔv] (Δv is Volterra’s Laplacian) is real analytic in t ∊ R as a one-parameter group of operators on ℋ, etc.

The purpose of this paper is to investigate a quasi-flow which is a one-parameter group of non-singular measurable point transformations on a measure space. If, in particular, the transformations are all measure preserving (i.e. a flow is given), the ergodicity together with the mixing property, the spectral or metrical type, increasing partitions of the space and the entropy of the flow are our main interests. Those methods used in the study of a flow are frequently useful for our approach. For example, the concept of a special flow introduced by W. Ambrose [3] plays an important role and the representation of a given flow in terms of a special flow is a powerful tool in the study of flows. L.M. Abramov [1] calculates the entropy of a flow with the help of the representation. As another example we give attention to the work of G. Maruyama [10] and H. Totoki [15] where they discuss a general time-change of flows the basic idea of which was originated by E. Hopf [8]. They discuss the invariant measure of a general time-changed flow and prove that the ergodicity is inherited and the entropy is kept invariant by the time-change. In the study of quasi-flows, we shall use both the repesentation in terms of a special quasi-flow and a time-change. Besides a quasi-flow requires its own methods in the investigation and it gives us some further problems such as the existence of an invariant measure (cf. [4], [5], [6] and [7]) and related topics.

In [1], G. Shimura gave a very practical formula of the traces of the Hecke operators acting on the space of cusp forms of rational weight and there he emphasized that the traces are effectively computable. We shall practice the computation in some special cases and discuss the structure of the Hecke algebra, which is not necessarily semi-simple.

By a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we examine the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First, we show that the sequence of heat kernels corresponding to the degenerating family converges, after rescaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff, the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called the complexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study of I-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz–type zeta functions.

It was shown in Eilenberg-Nagao-Nakayama [3] (Theorem 8 and § 4) that if Ω is an algebra (with unit element) over a field K with (Ω: K) <∞ and if the cohomolgical dimension of Ω, dim Ω, is ≦ 1, then every residue-algebra of Ω has a finite cohomological dimension. In the present note we prove a theorem of converse type, which gives, when combined with the cited result, a rather complete general picture of algebras whose residue-algebras are all of finite cohomological dimension.

The natural numbers play a part in the formulation of logical syntax inasmuch as they are used to count the symbols in a sentence, or the sentences in a proof, etc. In the present paper, we shall study an infinitary logical calculus which is based on replacing the ordinary natural numbers in the capacity just mentioned, by a non-standard model of arithmetic. (Compare refs. 3, 5, 6 for some other logical calculi of infinitary nature). Thus, our calculus will include formulae of length n for any natural number n, finite or infinite, in the chosen non-standard model of arithmetic. Evidently, the study of such formulae can be of interest only if we introduce concepts which are beyond the power of expression of the notions borrowed from the standard case. It turns out that the concept of truth in a model, when defined by means of Skolem functions has this character and involves a curious phenomenon which is analogous to one first pointed out by H. Steinhaus and J. Mycielski for another kind of infinitary language. Thus, while in the standard predicate calculus the negation of a sentence in prenex normal form is reduced to prenex normal form by changing the type of the quantifiers and by shifting the sign of negation, this procedure is not legitimate when truth is defined in this way in our calculus.

In the present note we wish to deal with the same problem as the preceding paper [1] for the case of modular fields.

Let k be a ñeiá of characteristic and K = k(x1 …, xp) a purely-transcendental extension of k, Let S be the automorphism of K which is induced by the cyclic permutation (x1., .., xp) and L the fixed subfield of S. Then L is a purely transcendental extension field over k.

Recently A, Malcev has shown that the homogeneous space of a connected nilpotent Lie group G is the direct product of a compact space and an Euclidean-space and that the compact space of this direct decomposition is also a homogeneous space of a connected subgroup of G. Any compact homogeneous space M of a connected nilpotent Lie group is of the form where is a connected simply connected nilpotent group whose structure constants are rational numbers in a suitable coordinate system and D is a discrete subgroup of G.