A Version of the Weak Krull–Schmidt Theorem for Infinite Direct Sums of Uniserial Modules

We show a version of the weak Krull–Schmidt theorem concerning infinite families of uniserial modules.     

The Krull―Schmidt Theorem for Henselian Rings

A necessary and sufficient condition on a local ring over which all indecomposable finite-dimensional algebras are local is found. The KrullSchmidt theorem for a class of rings that includes both the Henselian valuation rings and the rings of integers of multidimensional fields is proved. Bibliography: 2 titles.     

Krull–Remak–Schmidt Fails for Artinian Modules over Local Rings

Let R be a ring. Any R-module M which is Artinian or Noetherian can be written as the direct sum of a finite number of indecomposable R-modules. The theorem of Krull–Remak–Schmidt asserts that in the case where M is of finite length, such a decomposition is unique up to isomorphism. On the other hand, examples of Noetherian R-modules which have essentially different decompositions have been known for a long time. The first examples of Artinian R-modules with essentially different decompositions were published only in 1995 by Facchini, Herbera, Levy and Vámos. In order to construct such examples, one needs to deal with suitable rings R. Note that for R Noetherian or commutative, all the Artinian modules have the Krull–Remak–Schmidt property. In 1998, Facchini raised the problem of whether the same is true in the case where R is a local ring. The aim of this note is to show that this is not so: we are going to present a local ring R and Artinian R-modules M with essentially different direct decompositions into indecomposables. The military importance of these results has been discussed during the NATO meeting at Constantia (August 2000) which was organized by K. W. Roggenkamp.     

The Average Range Characterization of the Radon–Nikodym Property

A characterization of Banach spaces possessing the Radon—Nikodym property is given in terms of the average range of additive interval functions. We prove that a Banach space X has the RNP if and only if each X-valued additive interval function possessing absolutely continuous McShane (or Henstock) variational measure has nonempty average range almost everywhere on [0, 1].     

The Krull–Gabriel Dimension of a Serial Ring

Abstract

We prove that every serial ring R has the isolation property: every isolated point in any theory of modules over R is isolated by a minimal pair. Using this we calculate the Krull–Gabriel dimension of the module category over serial rings. For instance, we show that this dimension cannot be equal to 1.     

The Infinite Dimensional Hyperbolic Space H~∞ Does Not Have Property A

The author constructs a sequence of cubes in the infinitely dimensional hyperbolic space H∞ which is equi-coarsely equivalent to Z2n. As a corollary, it is proved that the infinitely dimensional hyperbolic space H∞ does not have property A.     

The Coordinatewise Uniformly Kadec–Klee Property in Some Banach Spaces

We introduce a new property UKK _c for a Banach space and show for an Orlicz sequence space the following: UKK _c H _{c 2}.     

The Property of Bair in r-Space

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Optimal Control for the Convective Cahn–Hilliard Equation in 2D Case

In this paper, for 2D convective Cahn–Hilliard equation, the optimal control problem is considered, the existence of optimal solution is proved and the optimality system is established.     

Adaptative Decomposition: The Case of the Drury–Arveson Space

The maximum selection principle allows to give expansions, in an adaptive way, of functions in the Hardy space \(\mathbf H_2\) of the disk in terms of Blaschke products. The expansion is specific to the given function. Blaschke factors and products have counterparts in the unit ball of \(\mathbb C^N\), and this fact allows us to extend in the present paper the maximum selection principle to the case of functions in the Drury–Arveson space of functions analytic in the unit ball of \(\mathbb C^N\). This will give rise to an algorithm which is a variation in this higher dimensional case of the greedy algorithm. We also introduce infinite Blaschke products in this setting and study their convergence.     

The Carnot-Carathéodory Distance and the Infinite Laplacian

In ℝ ⁿ equipped with the Euclidean metric, the distance from the origin is smooth and infinite harmonic everywhere except the origin. Using geodesics, we find a geometric characterization for when the distance from the origin in an arbitrary Carnot-Carathéodory space is a viscosity infinite harmonic function at a point outside the origin. We show that at points in the Heisenberg group and Grushin plane where this condition fails, the distance from the origin is not a viscosity infinite harmonic subsolution. In addition, the distance function is not a viscosity infinite harmonic supersolution at the origin.     

The K-Theory of Cuntz–Krieger Algebras for Infinite Matrices

We compute the K-theory groups of the Cuntz–Krieger C^*-algebra O_A associated to an infinite matrix A of zeros and ones.     

Infinite retrogression in the Eaves—Saigal algorithm

For certain functionsf fromR ⁿ toR ⁿ , the Eaves—Saigal algorithm computes a path inR ⁿ × (0, 1] which converges to a zero off. In this case, it is shown that even whenf is of classC and has a unique zero, the converging path may retrogress infinitely many times.Army Research Office, Durham, Contract No. DAAG-29-78-G-0026; National Science Foundation Grant No. MCS-77-05623.     

Graphs with the Strong Havel–Hakimi Property

The Havel–Hakimi algorithm iteratively reduces the degree sequence of a graph to a list of zeroes. As shown by Favaron, Mahéo, and Saclé, the number of zeroes produced, known as the residue, is a lower bound on the independence number of the graph. We say that a graph has the strong Havel–Hakimi property if in each of its induced subgraphs, deleting any vertex of maximum degree reduces the degree sequence in the same way that the Havel–Hakimi algorithm does. We characterize graphs having this property (which include all threshold and matrogenic graphs) in terms of minimal forbidden induced subgraphs. We further show that for these graphs the residue equals the independence number, and a natural greedy algorithm always produces a maximum independent set.     

The Bochner–Schoenberg–Eberlein Property for Totally Ordered Semigroup Algebras

The concepts of BSE property and BSE algebras were introduced and studied by Takahasi and Hatori in 1990 and later by Kaniuth and Ülger. This abbreviation refers to a famous theorem proved by Bochner and Schoenberg for \(L^1({\mathbb {R}})\), where \({\mathbb {R}}\) is the additive group of real numbers, and by Eberlein for \(L^1(G)\) of a locally compact abelian group G. In this paper we investigate this property for the Banach algebra \(L^p(S,\mu )\;(1\le p<\infty )\) where S is a compact totally ordered semigroup with multiplication \(xy=\max \{x,y\}\) and \(\mu \) is a regular bounded continuous measure on S. As an application, we have shown that \(L^1(S,\mu )\) is not an ideal in its second dual.     

Stationary Spherically Symmetric Solutions of the Vlasov–Poisson System in the Three-Dimensional Case

Doklady Mathematics - We consider the three-dimensional stationary Vlasov–Poisson system of equations with respect to the distribution function of the gravitating matter $$f =...