To a pair $P$ and $Q$ of finite posets we attach the toric ring $K[P,Q]$ whose generators are in bijection to the isotone maps from $P$ to $Q$. This class of algebras, called isotonian, are natural generalizations of the so-called Hibi rings. We determine the Krull dimension of these algebras and for particular classes of posets $P$ and $Q$ we show that $K[P,Q]$ is normal and that their defining ideal admits a quadratic Gröbner basis.

We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ring R of mixed characteristic p > 0, where p is a nonzero divisor, if I is an ideal of finite projective dimension over R and p 𝜖 I or p is a nonzero divisor on R/I, then every minimal generator of I is a nonzero divisor. Hence, if P is a prime ideal of finite projective dimension in a local ring R, then every minimal generator of P is a nonzero divisor in R.

We give an explicit construction of the Ree groups of type G2 as groups acting on mixed Moufang hexagons together with detailed proofs of the basic properties of these groups contained in the two fundamental papers of Tits on this subject (see [7] and [8]). We also give a short proof that the norm of a Ree group is anisotropic.

The purpose of this paper is to determine left-invariant vector fields on a Lie group G with a left-invariant Riemannian metric which induces C- flows on G.

In characteristic $0$ , symplectic automorphisms of K3 surfaces (i.e., automorphisms preserving the global $2$ -form) and non-symplectic ones behave differently. In this paper, we consider the actions of the group schemes $\mu _{n}$ on K3 surfaces (possibly with rational double point [RDP] singularities) in characteristic p, where n may be divisible by p. We introduce the notion of symplecticness of such actions, and we show that symplectic $\mu _{n}$ -actions have similar properties, such as possible orders, fixed loci, and quotients, to symplectic automorphisms of order n in characteristic $0$ . We also study local $\mu _n$ -actions on RDPs.

There are various generalizations of Hopf’s brilliant theorem, which may be stated, as newly formulated by Alexandroff; all the homotopy classes of the mappings of a compact Hausdorff space X with dim X≦n into an n-sphere Sn are in a (1-1) -correspondence with the elements of the n-dimensional Čech cohomology group Hn(X) with integer coefficients.

In his paper [2], [3], D. A. Hejhal investigated the variational theory of linear polynomic functions. In this paper we are concerned with the variational theory of higher-order differential equations. To be more precise, consider a compact Riemann surface having genus g > 1. As is well known, we can choose a projective coordinate covering U = (Ua, za). Fix this coordinate covering of X. We shall be concerned with linear ordinary differential operators of order n defined in each projective coordinate open set Ua

Let M = M(D π) be a (real analytic) fibered manifold of dimension n over a manifold D of dimension p, with projection π. We denote by M′ (D, α) the prolonged fibered manifold of M(D, π). Every point of M′ is a p-dimensional contact element of M, and a p-dimensional contact element X of M belongs to M′ if and only if π*X = Tπ(x)(D), where x is the origin of X. We write x = β(X) and α = π°β. We also denote by M″(D, α′) the prolonged fibered manifold of M′(D, α), where α′ = α°β′ and β′ is the projection of each X′ ∈ M″ to its origin in M′.

The device of representing a predicate by a universal sentence gives rise to logics of higher order. To represent an object constructed by a series of certain steps, we usually use terms. To represent a predicate constructed by a series of certain steps, we now use sentences in general. We can now substitute special predicates represented by sentences for predicate variables of a sentence just as we have been able to substitute special objects represented by terms for object variables. It is remarkable that we can always distinguish predicates from objects, proceeding in this way.

Dans toute la suite Rn désignera l’espace euclidien à dimension n(≧ 1). Pour un point x = x1, x2, … xn) de Rn, on note |x| = et la coordonnée sphérique dans Rn désignera (r, α).

Rappelons qu’un noyau de convolution N sur Rn est une mesure (de Radon) positive dans Rn dans la théorie du potentiel. Pour une mesure réelle μ dans Rn, N * μ s’appelle le N-potentiel de μ dès que cette convolution est définie au sens des mesures. On connaît bien que, dans la théorie du potentiel, les noyaux de convolution de Hunt possédent les propriétés définitives.

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).

We study geometry of manifolds endowed with a Grassmann structure which depends on symmetries of their curvature. Due to this reason a complete decomposition of the space of curvature tensors over tensor product vector spaces into simple modules under the action of the group G = GL(p, ℝ) ⊗ GL(q, ℝ) is given. The dimensions of the simple submodules, the highest weights and some projections are determined. New torsion-free connections on Grassmann manifolds apart from previously known examples are given. We use algebraic results to reveal obstructions to the existence of corresponding connections compatible with some type of normalizations and to enlighten previously known results from another point of view.

We are concerned with the question how the capacity of the ideal boundary of a subsurface of a covering Riemann surface over a Riemann surface varies according to the variation of its branch points. In the present paper we treat the most primitive but fundamental situation that the covering surface is a two sheeted sphere with two branch points one of which is fixed and the other is moving and the subsurface is given as the complement of two disjoint continua each in different sheets of the covering surface whose projections are two disjoint continua in the base plane given in advance not touching the projections of branch points. We will derive a variational formula for the capacity and as one of its many useful consequences expected we will show that the capacity changes smoothly as one branch point moves in the subsurface.

We study the asymptotic distribution of rational points on a generalized flag variety which are of bounded height and satisfy some congruence conditions in the formulation analogous to a strongly Hardy-Littlewood variety.

A deformation theory for rings and algebras was introduced recently by M. Gerstenhaber [1]. Let K be an extension of a field k, and p denotes the characteristic. One of his results is that, if K is separable over k, then it is rigid. It was conjectured in [1] that, if K is not separable over k, then it is not rigid, and if it is further finitely generated, then an integrable element of (see [2]) will be found in the image of Sqp. In this note we shall study the above conjecture in certain special case.

Let Q2 be the 2-adic number field, T/Q2 be a finite unramified extension, ζv be a primitive 2v-th root of unity, and let Kv = T(ζv). In a previous paper [1, Theorem 11], we stated the following theorem without its proof.

Let D denote the open unit disk in the complex plane and let C be the boundary of D. If, for a given complex-valued function f(z) defined in D, the existence of a subset M of C is known, with the linear measure of M equal to 2 π, as well as an estimate on the growth of |f/(z)| I on sequences in D which tends to a point of M, then such a result will be called a “statistical” result on order. This terminology is due to Lelong-Ferrand [3].

Unfortunately, there is a mistake in [PS, Lemma 3.10] which invalidates [PS, Theorem 3.12]. We show that the theorem still holds if the ring is assumed to be Gorenstein.

L. Carleson ([6]) proved the following theorem:

Let u be a finite continuous function in the unit open ball B with center zero in the complex plane.

Soient X un espace localement compact et non-compact à base dénombrable, G une noyau-fonction continue sur X et M (resp. MK) l’ensemble des toutes mesures de Radon positives sur X (resp. des toutes mesures de Radon positives sur X à support compact).