Decompositions of continuity and complete continuity

We introduce the notions of δ-t-sets, δ_β-t-sets, δ-B-continuity and δ_β -B-continuity and obtain decompositions of continuity and complete continuity.     

Decompositions of some weaker forms of continuity

The aim of this paper is to give decompositions of some weaker forms of continuity using the concepts of classes B ₁, B ₂, B ₃, αA and αC introduced by ourselves.     

Decompositions of rings

Translated from Matematicheskie Zametki, Vol. 57, No. 1, pp. 137–139, January, 1995.     

Decompositions of comodules

In this paper we study the relations among the different decomposition theories of comodules.     

Decompositions of degenerations

We develop a method to refine a given degeneration of modules. This enables us to give a new proof of the equivalence of the partial orders ≤_deg and ≤_ext in the case that the algebra Λ has a directed Auslander–Reiten quiver. Received: 11 November 1999     

Decompositions of semigroups

Communicated by Gerard J. Lallement     

Decompositions of signed-graphic matroids

We give a decomposition theorem for signed graphs whose frame matroids are binary and a decomposition theorem for signed graphs whose frame matroids are quaternary.     

External Paradoxical Decompositions

Ore's condition states that a cancellative semigroup S which has common right multiples embeds into a group G such that certain properties are satisfied by S and G. We show that G is nonamenable if and only if the semigroup S^-1 is G-paradoxical with respect to right multiplication by elements of S. We explore certain properties of this decomposition of S^-1.     

Sparsity-certifying Graph Decompositions

We describe a new algorithm, the (k, ℓ)-pebble game with colors, and use it to obtain a characterization of the family of (k, ℓ)-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the (k, ℓ)-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow [5], Gabow and Westermann [6] and Hendrickson [9]. Ileana Streinu; Research of both authors funded by the NSF under grants NSF CCF-0430990 and NSF-DARPA CARGO CCR-0310661 to the first author.     

Fundamental Agler Decompositions

We use shift-invariant subspaces of the Hardy space on the bidisk to provide an elementary proof of the existence of Agler decompositions. We observe that these shift-invariant subspaces are specific cases of Hilbert spaces that can be defined from Agler decompositions and analyze the properties of such Hilbert spaces. We then restrict attention to rational inner functions and show that the shift-invariant subspaces provide easy proofs of several known results about decompositions of rational inner functions. We use our analysis to obtain a result about stable polynomials on the polydisk.     

Induced Decompositions of Graphs

We consider those graphs G that admit decompositions into copies of a fixed graph F, each copy being an induced subgraph of G. We are interested in finding the extremal graphs with this property, that is, those graphs G on n vertices with the maximum possible number of edges. We discuss the cases where F is a complete equipartite graph, a cycle, a star, or a graph on at most four vertices.     

Decompositions of binomial ideals

We present Binomials, a package for the computer algebra system Macaulay 2, which specializes well-known algorithms to binomial ideals. These come up frequently in algebraic statistics and commutative algebra, and it is shown that significant speedup of computations like primary decomposition is possible. While central parts of the implemented algorithms go back to a paper of Eisenbud and Sturmfels, we also discuss a new algorithm for computing the minimal primes of a binomial ideal. All decompositions make significant use of combinatorial structure found in binomial ideals, and to demonstrate the power of this approach we show how Binomials was used to compute primary decompositions of commuting birth and death ideals of Evans et al., yielding a counterexample for their conjectures.     

Decompositions of determinantal varieties

In this paper, we study the irreducible decompositions of determinantal varieties of matrices given by rank conditions on upper left submatrices. Using the concept of essential rank function and the Ehresmann partial order on the set of all simple matrices, we design an algorithm to write a determinantal variety as a union of its irreducible components. This solves a problem raised by B. Sturmfels.      

Decompositions of Complete Graphs

If s₁, s₂, ..., s_t are integers such that n – 1 = s₁ +s₂ + ... + s_t and such that for each i (1 i t), 2 s_i n –1 and s_in is even, then K_n can be expressed as the union G₁G₂...G_tof t edge-disjoint factors, where for each i, G_i is s_i-regularand s_i-connected. Moreover, whenever s_i = s_j, G_i and G_j areisomorphic. 1991 Mathematics Subject Classification 05C70.     

Star Decompositions of Cubes

 Let S _k denote the complete bipartite graph K _{1, k} and let Q _n denote the n-cube. We prove that the obvious necessary conditions for the existence of an S _k -decomposition of Q _n are sufficient. Received: July 21, 1999 Final version received: May 16, 2000