In this paper we are interested in spectral decomposition of an unbounded operator with discrete spectrum. We show that if A generates a polynomially bounded n-times integrated group whose spectrum set σ(A) = {iλk; k ∈ Z} is discrete and satisfies ∑ 1 |λk|δ k < ∞ (n and l nonnegative integers), then there exists projectors (Pk)k∈Z∗ such that ∑ Pkx = x (x ∈ D(A)), where δk = min ( |λk+1−λk| 2 , ...

The following theorem is due to Dilworth [12]: Let P be a partially ordered set. If the maximal number of elements in an independent subset (anti-chain) of P is k, then P is the union of k chains (cliques). In this article we formalize an elegant proof of the above theorem for finite posets by Perles [17]. The result is then used in proving the case of infinite posets following the original pro...

DECOMPOSITION THEOREM AND APPLICATIONS RAMI GROSSBERG AND OLIVIER LESSMANN Abstract. In this paper, we prove a decomposition theorem for abstract elementary classes K with the amalgamation property, under the assumption that certain axioms regarding independence, existence of some prime models, and regular types are satisfied. This context encompasses the following: (1) K is the class of models...

Fix c : [N] → r, an r-coloring of the pairs of natural numbers. An ordered list of distinct integers, a0, a1, a2 . . . ai−1, ai, ai+1 . . . is a monochromatic path for color k, if, for all i ≥ 1, c({ai−1, ai}) = k. The empty list is considered a path of any color k. Similarly the list of one element, a0, is also considered a path of any color k. For any monochromatic path of length two or more ...

Table 2: Percent a b o ve optimum of Elmore delay to a single critical sink and wire length for three Steiner tree constructions cost comparison is with 1-Steiner. Averages were taken over 200 random nets for each net size. 4.2 Elmore-Optimality of Generic" SERT Algorithm The counterexample in Section 3.3 showing that BB-SORT is not always optimal was carefully constructed by hand; even then, B...

In this note we point out the relation between Brion’s formula for the lattice point generating function of a convex polytope in terms of the vertex cones [Bri88] on the one hand, and the polar decomposition à la Lawrence/Varchenko [Law91a, Var87] on the other. We then go on to prove a version of polar decomposition for non-simple polytopes.

A domain constructor that generalizes the product is de ned. It is shown that with this constructor exactly the prime-algebraic coherent Scott-domains and the empty set can be generated from two-chains and boolean at domains. 3 List of Symbols I am identifying the symbols by the corresponding Latex(+Amssymb)-symbols. " uparrow # downarrow ! rightarrow ? bot > top leq geq 2 in W bigvee V bigwedg...

We give a decomposition theorem for Platonic graphs over finite fields and use this to determine the spectrum of these graphs. We also derive estimates for the isoperimetric numbers of the graphs. THE SPECTRUM OF PLATONIC GRAPHS OVER FINITE FIELDS MICHELLE DEDEO, DOMINIC LANPHIER, AND MARVIN MINEI Abstract. We give a decomposition theorem for Platonic graphs over finite fields and use this to d...

1 OPERATOR DECOMPOSITIONS In this lecture we will consider two basic decompositions of operators that will be used repeatedly in the course: the spectral decomposition and the singular-value decomposition. The spectral decomposition holds only for normal operators, while the singular-value decomposition holds for all operators. The singular-value decomposition will be particularly important in ...

We show that all the tangles in a finite graph or matroid can be distinguished by a single tree-decomposition that is invariant under the automorphisms of the graph or matroid. This comes as a corollary of a similar decomposition theorem for more general combinatorial structures, which has further applications. These include a new approach to cluster analysis and image segmentation. As another ...