The present paper constitutes a continuation of the ideas and methods of M. Morse and M. Heins [1]. As in that work the subject treated is the theory of deformation classes of meromorphic functions and interior transformations. There the functions considered were defined over the open disc < 1 and had only a finite number of zeros, poles and branch point antecedents. It is possible to transfer the results obtained to the situation where the domain of definition is any simply-connected domain of hyperbolic type or, alternatively, of parabolic type.

Pour toute sous-vari ét é géométriquement irréductible $V$ du groupe multiplicatif $\mathbb{G}_{m}^{n}$, on sait qu’en dehors d’un nombre fini de translat és de tores exceptionnels inclus dans $V$, tous les points sont de hauteur minorée par une certaine quantité $q{{(V)}^{-1}}>0$. On connaît de plus une borne supérieure pour la somme des degrés de ces translatés de tores pour des valeurs de $q(V)$ polynomiales en le degré de $V$. Ceci n’est pas le cas si l’on exige une minoration quasi-optimale pour la hauteur des points de $V$, essentiellement linéaire en l’inverse du degré.

Nous apportons ici une réponse partielle à ce problème : nous donnons une majoration de la somme des degrés de ces translat és de sous-tores de codimension 1 d’une hypersurface $V$. Les résultats, obtenus dans le cas de $\mathbb{G}_{m}^{3}$,mais complètement explicites, peuvent toutefois s’étendre à $\mathbb{G}_{m}^{n}$,moyennant quelques petites complications inhérentes à la dimension $n$.

Until recently the literature contained little on the axiomatic foundations of elliptic geometry that was non-analytical and independent of projective geometry. During the past decade this subject has come in for further study, notably by Busemann [2] and Blumenthal [1], who supplied such foundations. This paper presents another and, it is believed, simpler effort in the same general direction, proceeding by the familiar synthetic methods of elementary geometry and using only elementary topological notions and ideas concerning metric spaces. Specifically, elliptic 2-space is obtained on the basis of six axioms, most notable of which is one assuming the existence of translations. The writer wishes to express his deep appreciation to Herbert Busemann for his invaluable help.

We investigate the pluriclosed flow on Oeljeklaus–Toma manifolds. We parameterize left-invariant pluriclosed metrics on Oeljeklaus–Toma manifolds, and we classify the ones which lift to an algebraic soliton of the pluriclosed flow on the universal covering. We further show that the pluriclosed flow starting from a left-invariant pluriclosed metric has a long-time solution $\omega _t$ which once normalized collapses to a torus in the Gromov–Hausdorff sense. Moreover, the lift of $\tfrac {1}{1+t}\omega _t$ to the universal covering of the manifold converges in the Cheeger–Gromov sense to $(\mathbb H^s\times \mathbb C^s, \tilde {\omega }_{\infty })$, where $\tilde {\omega }_{\infty }$ is an algebraic soliton.

In this paper we prove that the order of the collineation group of the Veblen-Wedderburn plane of order nine is 311,040. This result was stated by Hall [3] in 1943 and proved by Pierce [9] in 1964. Hall assumed that there were 10 · 8 · 6 · 4 · 2 = 3840 collineations which permute points on the ideal line L and 81 collineations which leave L pointwise fixed. In 1955 André [1] verified this assumption. When it was realized that a harmonic homology with axis L had been overlooked, the number of central collineations with axis L doubled and hence the order of the collineation group became 3840 · 162 = 622,080. This latter figure has been assumed to be correct as recently as 1965 ([6]).

Here it is proved that there are 1920 collineations which move points on L and 162 collineations which leave L pointwise fixed, thus giving the figure 311,040. Pierce's proof of this fact is established from a different viewpoint.

Although the behaviour of Newtonian potentials inside n-dimensional distributions of mass or charge has been discussed in the sense of Lebesgue-Stieltjes integrals by various authors, the discussion of various important theorems seems to have been made only in the sense of Riemann integration, and assuming the Hälder conditions (or at least piecewise continuity) for the volume density p. We shall generalize these theorems below.

Given a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic ${{L}^{2}}$ torsion, which lies in the determinant line of the twisted ${{L}^{2}}$ Dolbeault cohomology and represents a volume element there. Here we utilise the theory of determinant lines of Hilbertian modules over finite von Neumann algebras as developed in $[\text{CFM}]$. This specialises to the Ray-Singer-Quillen holomorphic torsion in the finite dimensional case. We compute a metric variation formula for the holomorphic ${{L}^{2}}$ torsion, which shows that it is not in general independent of the choice of Hermitian metrics on the complex manifold and on the holomorphic Hilbertian bundle, which are needed to define it. We therefore initiate the theory of correspondences of determinant lines, that enables us to define a relative holomorphic ${{L}^{2}}$ torsion for a pair of flat Hilbertian bundles, which we prove is independent of the choice of Hermitian metrics on the complex manifold and on the flat Hilbertian bundles.

First, we build a computational procedure to reconstruct the convex body from a pre-given surface area measure. Nontrivially, we prove the convergence of this procedure. Then, the sufficient and necessary conditions of a Sobolev binary function to be a lightness function of a convex body are investigated. Finally, we present a computational procedure to compute the curvature function from a pre-given lightness function, and then we reconstruct the convex body from this curvature function by using the first procedure. The convergence is also proved. The main computations in both procedures are simply about the spherical harmonics.

1. The Theorem of Le Roux. Let U(x, y, α) be any solution of the linear hyperbolic differential equation

1

containing a parameter α. J. Le Roux (5) has shown that the function u(x, y) defined by

2 α0 = const.

A theorem of Eisenbud, Griffith, and Robson states that if R is hereditary and noetherian (on both the left and right) then every proper homomorphic image of R is a generalized unserial ring (see, for example, [3, p. 244]). Singh [11, p. 883] states a converse: If R is a right bounded, noetherian prime ring, all of whose proper homomorphic images are generalized uniserial rings, then (every divisible right R-module is injective, so) R is right hereditary. (Actually, Singh omitted the clearly necessary “bounded” condition.) Singh's theorem generalizes results of [9, Proposition 15], [2, Theorem 2.1], and [8], about commutative rings.

We will call a semi-prime ring R essentially right bounded if each essential right ideal contains a two-sided ideal which is essential as a right ideal. In case R is prime, “essentially right bounded” coincides with “right bounded”.

A remarkable theorem of Herstein [1, Theorem 2] of which we have made several uses states: If R is a semiprime ring of characteristic different from 2 and if U is both a Lie ideal and a subring of R then either U ⊂ Z (the centre of R) or U contains a nonzero ideal of R. In a recent paper [3] Herstein extends the above mentioned result significantly and has proved that if R is a semiprime ring of characteristic different from 2 and V is an additive subgroup of R such that [V, U] ⊂ V, where U is a Lie ideal of R, then either [V, U] = 0 or V ⊃ [M, R] ≠ 0 where M is an ideal of R. In this paper our object is to prove the following.

The density u(x, t) of an ideal gas flowing through a homogeneous porous media satisfies the equation Here m > 1 is a physical constant and u also satisfies the initial condition If the initial data is not strictly positive it is necessary to work with generalized solutions of the Cauchy problem (1), (2) (see [1]). By a weak solution we shall mean a function u(x, t) such that for (in the sense of distributions) and

Let $M$ be the product of two compact Hamiltonian $T$-spaces $X$ and $Y$. We present a formula for evaluating integrals on the symplectic reduction of $M$ by the diagonal $T$ action. At every regular value of the moment map for $X\,\times \,Y$, the integral is the convolution of two distributions associated to the symplectic reductions of $X$ by $T$ and of $Y$ by $T$. Several examples illustrate the computational strength of this relationship. We also prove a linear analogue which can be used to find cohomology pairings on toric orbifolds.

Let $\phi :\,X\,\to \,M$ be a morphism from a smooth irreducible complex quasi-projective variety $X$ to a Grassmannian variety $M$ such that the image is of dimension ≥ 2. Let $D$ be a reduced hypersurface in $M$, and $\gamma $ a general linear automorphism of $M$. We show that, under a certain differential-geometric condition on $\phi (X)$ and $D$, the fundamental group ${{\text{ }\!\!\pi\!\!\text{ }}_{1}}\left( {{\left( \gamma \,o\,\phi \right)}^{-1}}\,\left( M\,\backslash \,D \right) \right)$ is isomorphic to a central extension of ${{\pi }_{1}}\left( M\,\backslash \,D \right)\,\,\times \,{{\pi }_{1}}\left( X \right)$ by the cokernel of ${{\pi }_{2}}\left( \phi \right)\,:\,{{\pi }_{2}}\left( X \right)\,\to {{\pi }_{2}}\left( M \right)$.

A Banach space B is called an isometric predual, or simply a predual, of a Banach space X if the dual B* of B is isometrically isomorphic to X. A Banach space X is said to have a unique (isometric) predual if X has a predual and all preduals are mutually isometrically isomorphic. In general a Banach space does not have a unique predual even if it has a predual. A simple example of this is the space l1, because c0 and c are isometric preduals of l1 but not isometrically isomorphic. A. Grothendieck [3] first noticed that L∞-spaces have unique preduals, and then S. Sakai generalized this to von Neumann algebras (see p. 30 of [9]).

In this paper I establish the following bounding formula for the integral of a function of n variables:

(1.1)

Let D be a bounded domain in the Euclidean space RN(N ≦ 2) and let where is the closure of D. We assume that the boundary ∂G of G is smooth. Consider the boundary value problem (abbreviated to BVP in the sequel).

(1)

(1)

where A is a nonlinear elliptic differential operator in divergence form of Leray-Lions type, ∇u = grad u, f is a distribution on G and p(x, t, η) is a function defined on G × R × RN. Among other hypotheses we shall, roughly speaking, assume that p has completely unrelated growth rates in the first and the third variables. In this paper we prove the solvability of the BVP (1), (2) under the assumption that it has both an upper solution ψ and a lower solution φ with φ ≧ ψ.

In a recent paper [6], P. Seibt has obtained the following result: Let k be a field of characteristic 0, k[T1, … , Tr] the polynomial ring in r indeterminates over k, and let P be a prime ideal of k[T1, … , Tr]. Then a polynomial F belongs to the n-th symbolic power P(n) of P if and only if all higher derivatives of F from the 0-th up to the (n – l)-st order belong to P.

In this work we shall naturally generalize this result so as to be valid for primes of the polynomial ring over a perfect field k. Actually, we shall get a generalization as a corollary to a theorem which asserts: For regular primes P in a k-algebra R of finite type, a certain differential filtration of R associated with P coincides with the symbolic power filtration (P(n))n≧0.

Let T be an invertible measure-preserving transformation on a σ-finite measure space (X, μ) and let 1 < p < ∞. This paper uses an abstract method developed by José Luis Rubio de Francia which allows us to give a unified approach to the problems of characterizing the positive measurable functions v such that the limit of the ergodic averages or the ergodic Hilbert transform exist for all f ∈ Lp(νdμ). As a corollary, we obtain that both problems are equivalent, extending to this setting some results of R. Jajte, I. Berkson, J. Bourgain and A. Gillespie. We do not assume the boundedness of the operator Tf(x) = f(Tx) on Lp(νdμ). However, the method of Rubio de Francia shows that the problems of convergence are equivalent to the existence of some measurable positive function u such that the ergodic maximal operator and the ergodic Hilbert transform are bounded from LP(νdμ) into LP(udμ). We also study and solve the dual problem.

In this paper we represent certain linear operators in a space with indefinite metric. Such a space may be a pair (H, B), where H is a separable Hilbert space, B is a bilinear functional on H given by B(x, y) = [Jx, y], [, ] is the Hilbert inner product in H, and J is a bounded linear operator such that J = J* and J2 = I. If T is a linear operator in H, then ‖T‖ is the usual operator norm. The operator J above has two eigenspaces corresponding to the eigenvalues + 1 and –1.

In case the eigenspace in which J induces a positive operator has finite dimension k, a general spectral theory is known and has been developed principally by Pontrjagin [25], Iohvidov and Kreĭn [13], Naĭmark [20], and others.