Let (R, m) be a local noetherian ring and I a proper ideal in R. Let (I) be the Rees-ring ⊕n≥0 In with respect to I. In this note we describe conditions for I and R in order that the Cohen-Macaulay property (C-M for short) of R/I can be lifted to R and (I), see Propositions 1.2, 1.3. and 1.4.

In this paper we define and study ladder determinantal rings of a symmetric matrix of indeterminates. We show that they are Cohen-Macaulay domains. We give a combinatorial characterization of their h-vectors and we compute the a-invariant of the classical determinantal rings of a symmetric matrix of indeterminates.

A nonnegative locally Hölder continuous function P(z) on 0 < | z | ≤ 1 will be referred to as a density on 0 < | z | ≤ 1. The elliptic dimension of a density P(z) at z = 0, dim P in notation, is defined to be the dimension of the half module of nonnegative solutions of the equation Δu(z) = P(z)u(z) on the punctured unit disk Ω : 0 < | z | < 1 with boundary values zero on | z | = 1. After Bouligand we say that the Picard principle is valid for a density P at z = 0 if dim P = 1.

Let be an infinitesimal group scheme, defined over an algebraically closed field of characteristic p. We employ rank varieties of -modules to study the stable Auslander-Reiten quiver of the distribution algebra of . As in case of finite groups, the tree classes of the AR-components are finite or infinite Dynkin diagrams, or Euclidean diagrams. We classify the components of finite and Euclidean type in case is supersolvable or a Frobenius kernel of a smooth, reductive group.

Let V be a complex vector space of dimension n, L a lattice of V and E = V/L a complex torus. Let H be a Hermitian form on V. We introduce a multiplication in L × C* by

Let us consider transporting particle in the n-dimensional Euclidian space Rn. It is assumed that a particle originating at a point x∈Rn moves in a straight line with constant speed c and continues to move until it suffers a collision. The probability that the particle has a collision between t and t + Δ is kΔ + o(Δ), where k is constant. When a particle has a collision, say at y in Rn, it moves afresh from y with an isotropic choice of direction independent of past history.

We investigate the uniform perfectness on a boundary point of a hyperbolic open set and distortion of a holomorphic function from the unit disk Δ into a hyperbolic domain with a uniformly perfect boundary point, especially of a universal covering map of such a domain from Δ, and we obtain similar results to celebrated Koebe’s Theorems on univalent functions.

It is well known that the concept of energy has played a fruitful role in potential theory and Markov processes. Cartan’s work [6] led to kernel-free potential theories of Beurling-Deny [2]. Since then many authors have worked on this, M. Fukushima [8], M. Silverstein [16], J. Bliedner [3], Berg-Forst [1], to name some. In these works, however, the main thrust is Hubert space theoretic.

The observation by Poincaré that Möbius transformations in the complex plane can be lifted to a half-space raises the need to be able to handle motions in hyperbolic space of more than two dimensions by means of an analytic apparatus of not too forbidding complexity. In my experience the best way to do so is to be guided by analogies with the familiar twodimensional case. The purpose of this little paper is to collect a few formulas that the writer has found useful when working with certain hyperbolically invariant operators.

Entropy of categorical dynamics is defined by Dimitrov–Haiden–Katzarkov–Kontsevich. Motivated by the fundamental theorem of the topological entropy due to Gromov–Yomdin, it is natural to ask an equality between the entropy and the spectral radius of induced morphisms on the numerical Grothendieck group. In this paper, we add two results on this equality: the lower bound in a general setting and the equality for orbifold projective lines.

Local densities of quadratic forms are important invariants in the theory of quadratic forms and they appear in Fourier coefficients of Eisenstein series. But it is not easy to evaluate them. To study their properties, it is desirable to look for relations among them, and it is known that there are many relations [3], but they are not concise. We consider a different kind of relations here and improve a result of Zharkovskaja [8, 9] in the case of Eisenstein series as an application.

Consider a nonnegative Hölder continuous 2-form P(z)dxdy (z = x + iy) on a connected Riemann surface R. We denote by P(R) the linear space of solutions u of the equation Δu = Pu on R and by PX(R) the subspace of P(R) consisting of those u with a certain boundedness property X. We also use the standard notations H(R) and HX(R) for P(R) and PX(R) with P ≡ 0.

We classify smooth Fano 4-folds with second betti number b2 = 2, possesing adjunction theoretic scroll structure. These manifolds occur to be projectivisations of coherent sheaves over Fano manifolds. The paper deals mainly with projectivisations of non-locally free rank 2 Fano sheaves over Fano 3-folds with b2 = 1, using Mori theory. By the way, we classify nef and big rank 2 bundles over P2 with the first Chern class (3).

Introduction and Preliminaries. Nakayama [9] suggested that algebras be classified according to the length of a right projective, injective resolution of the algebra as a left module. Tachikawa [10] showed that an algebra is QF-3 (see [12] for definition) if and only if the length is at least one. He [11] also gave some estimations of the length of the resolution besides relating the length to the size of the double commutator algebra of a faithful projective, injective left ideal.

1. Let Ω be a finite set of arbitrary elements and let (G, Ω) be a permutation group on Ω. (This is also simply denoted by G). Two permutation groups (G, Ω) and (G, Γ) are called isomorphic if there exist an isomorphism σ of G onto H and a one to one mapping τ of Ω onto Γ such that (g(i))τ=gσ(iτ) for g ∊ G and i∊Ω. For a subset Δ of Ω, those elements of G which leave each point of Δ individually fixed form a subgroup GΔ of G which is called a stabilizer of Δ. A subset Γ of Ω is called an orbit of GΔ if Γ is a minimal set on which each element of G induces a permutation. A permutation group (G, Ω) is called a group of rank n if G is transitive on Ω and the number of orbits of a stabilizer Ga of a ∊ Ω, is n. A group of rank 2 is nothing but a doubly transitive group and there exist a few results on structure of groups of rank 3 (cf. H. Wielandt [6], D. G. Higman M).

Let X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring Op,x at a rational singular point P of X, we attached a universal zeta function which is a rational function and admits a functional equation if Op,x is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.

The concept of a local ring was introduced by Krull [2], who defined it as a Noetherian ring R (we say that a commutative ring R is Noetherian if every ideal in R has a finite basis and if R contains the identity) which has only one maximal ideal m. If the powers of m are defined as a system of neighbourhoods of zero, then R becomes a topological ring satisfying the first axiom of countability, And the notion was studied recently by C. Chevalley and I. S. Cohen. Cohen [1] proved the structure theorem for complete rings besides other properties of local rings.

The (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps reachable indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is a categorification of the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a Serre functor and depends only on a rigid object in the category. It is well known that the usual Caldero–Chapoton map gives rise to so-called friezes, for instance, Conway–Coxeter friezes. We show that the modified Caldero–Chapoton map gives rise to what we call generalized friezes and that, for cluster categories of Dynkin type A, it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.

It is proved that every holomorphically convex complex space endowed with an action of a compact Lie group K can be realized as an open K-stable subspace of a holomorphically convex space endowed with a holomorphic action of the complexified group KC. Similar results are obtained for holomorphic if-bundles over such spaces.

We construct a bounded plane domain which is Bergman complete but for which the Bergman kernel does not tend to infinity as the point approaches the boundary.