Operator ergodic theory for one-parameter decomposable groups

After initial treatment of the Fourier analysis and operator ergodic theory of strongly continuous decomposable one-parameter groups of operators in the Banach space setting, we show that in the setting of a super-reflexive Banach space X these groups automatically transfer from the setting of R to X the behavior of the Hilbert kernel, as well as the Fourier multiplier actions of functions of higher variation on R. These considerations furnish one-parameter groups with counterparts for the single operator theory in Berkson (2010) [4]. Since no uniform boundedness of one-parameter groups of operators is generally assumed in the present article, its results for the super-reflexive space setting go well beyond the theory of uniformly bounded one-parameter groups on UMD spaces (which was developed in Berkson et al., 1986 [13]), and in the process they expand the scope of vector-valued transference to encompass a genre of representations of R that are not uniformly bounded.     

Roots of almost decomposable operators

The author shows that, for an injective analytic function f, f(T) is almost decomposable iff T is almost decomposable, where T is a bounded linear operator on a Banach space and f(T) is defined by the functional calculus.     

Finitely decomposable Banach spaces and the three-space property

A Banach space is hereditarily finitely decomposable if it does not contain finite direct sums of infinite dimensional subspaces with arbitrarily large number of summands. Here we show that the class of all hereditarily finitely decomposable Banach spaces has the three-space property. Moreover we show that the corresponding class defined in terms of quotients has also the three-space property.     

A new view toward vertex decomposable graphs

In this paper, we bring a new view about closed neighbourhood to show the vertex decomposability of graphs. Making use of the characterization of hereditary vertex decomposable graphs, we introduce a class of vertex decomposable graphs, which include some well-known classic vertex decomposable graphs such as clique-whiskered graphs and Cameron-Walker graphs.     

On freely decomposable maps

Freely decomposable and strongly freely decomposable maps were introduced by G.R. Gordh and C.B. Hughes as a generalization of monotone maps with the property that these maps preserve local connectedness in inverse limits. We study further these types of maps, generalize some of the results by Gordh and Hughes and present examples showing that no further generalization is possible.     

The duals of almost completely decomposable groups

In this paper, we classify the direct products of one-dimensional compact connected abelian groups by cardinal invariants dualizing Baer’s classification theorem of completely decomposable groups. Almost completely decomposable groups are finite rank torsion-free abelian groups which contain a completely decomposable group of finite index. An isomorphism theorem for their Pontrjagin dual groups is given by using the dual concept of a regulating subgroup.     

Algebraic shifting of strongly edge decomposable spheres

Recently, Nevo introduced the notion of strongly edge decomposable spheres. In this paper, we characterize algebraic shifted complexes of those spheres. Algebraically, this result yields the characterization of the generic initial ideal of the Stanley-Reisner ideal of Gorenstein^∗ complexes having the strong Lefschetz property in characteristic 0.     

From decomposable to residual theories

Over the last decade, first-order constraints have been efficiently used in the artificial intelligence world to model many kinds of complex problems such as: scheduling, resource allocation, computer graphics and bio-informatics. Recently, a new property called decomposability has been introduced and many first-order theories have been proved to be decomposable: finite or infinite trees, rational and real numbers, linear dense order, etc. A decision procedure in the form of five rewriting rules has also been developed. This latter can decide if a first-order formula without free variables is true or not in any decomposable theory. Unfortunately, the definition of decomposable theories is too much complex: theoretical and definitively not intuitive. As a consequence, checking whether a given theory T is decomposable is almost impossible for a non expert in decomposability. We introduce in this paper residual theories: a new class of first-order theories whose definition is very intuitive, easy to check and much more adapted to the needs of the artificial intelligence community. We show that decomposable theories is a sub-class of residual theories and present, not only a decision procedure, but a full first-order constraint solver for residual theories. It transforms any first-order constraint φ (which can possibly contain free variables) into an equivalent formula ? which is either the formula true, or the formula false or a simple solved formula having at least one free variable and being equivalent neither to true nor to false. We show the efficiency of our solver by solving complex first-order constraints containing long nested alternations of quantifiers over different residual theories.     

Recognizing decomposable graphs

A graph is called decomposable if its vertices can be colored red and blue in such a way that each color appears on at least one vertex but each vertex v has at most one neighbor having a different color from v. We point out a simple and efficient algorithm for recognizing decomposable graphs with maximum degree at most 3 and prove that recognizing decomposable graphs with maximum degree 4 is an NP-complete problem.     

Commutative Banach algebras and decomposable operators

For a multiplication operator on a semi-simple commutative Banach algebra, it is shown that the decomposability in the sense of Foia is equivalent to weak and to super-decomposability. Moreover, it can also be characterized by a convenient continuity condition for the Gelfand transform on the spectrum of the underlying Banach algebra. This result implies various permanence properties for decomposable multiplication operators and leads also to a useful characterization of the regularity for a semi-simple commutative Banach algebra. Finally, the greatest regular closed subalgebra of a commutative Banach algebra is investigated, and some applications to decomposable convolution operators on locally compact abelian groups are given.Research partially supported by NSF Grant DMS 90-96108.     

Robust decomposable Markov decision processes motivated by allocating school budgets

Motivated by an application to school funding, we introduce the notion of a robust decomposable Markov decision process (MDP). A robust decomposable MDP model applies to situations where several MDPs, with the transition probabilities in each only known through an uncertainty set, are coupled together by joint resource constraints. Robust decomposable MDPs are different than both decomposable MDPs, and robust MDPs and cannot be solved by a direct application of the solution methods from either of those areas. In fact, to the best of our knowledge, there is no known method to tractably compute optimal policies in robust, decomposable MDPs. We show how to tractably compute good policies for this model, and apply the derived method to a stylized school funding example.     

Osculating properties of decomposable scrolls

Osculating spaces of decomposable scrolls (of any genus and not necessarily normal) are studied and their inflectional loci are related to those of their generating curves by using systematically an idea introduced by Piene and Sacchiero in the setting of rational normal scrolls. In this broader setting the extra components of the second discriminant locus – deriving from flexes – are investigated and a new class of uninflected surface scrolls is presented and characterized. Further properties related to osculation are discussed for (not necessarily decomposable) scrolls (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A generalized decomposable multiattribute utility function

In this paper a generalized decomposable multiattribute utility function (MAUF) is developed. It is demonstrated that this new MAUF structure is more general than other well-known MAUF structures, such as additive, multiplicative, and multilinear. Therefore, it is more flexible and does not require that the decision maker be consistent with restrictive assumptions such as preferential independence conditions about his/her preferences. We demonstrate that this structure does not require any underlying assumption and hence solves the interdependence among attributes. Hence there is no need for verification of its structure. Several useful extensions and properties for this generalized decomposable MAUF are developed which simplify its structure or assessment. The concept of utility efficiency is developed to identify efficient alternatives when there exists partial information on the scaling constants of an assumed MAUF. It is assumed that the structure (decomposition) of the MAUF is known and the partial information about the scaling constants of the decision maker is in the form of bounds or constraints. For the generalized decomposable structure, linear programming is sufficient to solve all ensuing problems. Some examples are provided.     

A degree bound on decomposable trees

A n-vertex graph is said to be decomposable if for any partition (λ₁,…,λ_p) of the integer n, there exists a sequence (V₁,…,V_p) of connected vertex-disjoint subgraphs with |V_i|=λ_i. In this paper, we focus on decomposable trees. We show that a decomposable tree has degree at most 4. Moreover, each degree-4 vertex of a decomposable tree is adjacent to a leaf. This leads to a polynomial time algorithm to decide if a multipode (a tree with only one vertex of degree greater than 2) is decomposable. We also exhibit two families of decomposable trees: arbitrary large trees with one vertex of degree 4, and trees with an arbitrary number of degree-3 vertices.     

On the spatial distribution of solutions of decomposable form equations

We study the distribution in space of the integral solutions to an integral decomposable form equation, by considering the images of these solutions under central projection onto a unit ball. If we think of the solutions as stars in the night sky, we ask what constellations are visible from the earth (the unit ball). Answers are given for a large class of examples which are then illustrated using the software packages KANT and Maple. These pictures highlight the accuracy of our predictions and arouse interest in cases not covered by our results. Within the range of applicability of our results lie solutions to norm form equations and units in abelian group rings. Thus our theory has a lot to say about where these interesting objects can be found and what they look like.

     

Absolutely decomposable groups

We establish numerous results concerning the construction of decomposable and absolutely decomposable groups.     

Scott Brown's techniques for perturbations of decomposable operators

Using Scott Brown's techniques, J. Eschmeier and B. Prunaru showed that if T is the restriction of a decomposable (or S-decomposable) operator B to an invariant subspace such that (T) is dominating in C/S for some closed set S, then T has an invariant subspace. In the present paper we prove various invariant subspace theorems by weakening the decomposability condition on B and strengthening the thickness condition on (T).The research is supported by a grant from the Institute for Studies in Theoretical Physics and Mathematics (IRAN).     

Existence and relaxation theorems for extreme continuous selectors of multifunctions with decomposable values

We present some extreme continuous selector theorems, synthesizing the author's results; namely, we study existence and properties of continuous selectors from the set of extreme points of multifunctions with closed convex decomposable values in the space of Bochner integrable functions.