Optimality of the quantified Ingham–Karamata theorem for operator semigroups with general resolvent growth

We prove that a general version of the quantified Ingham–Karamata theorem for $$C_0$$C0-semigroups is sharp under mild conditions on the resolvent growth, thus generalising the results contained in a recent paper by the same authors. It follows in particular that the well-known Batty–Duyckaerts theorem is optimal even for bounded $$C_0$$C0-semigroups whose generator has subpolynomial resolvent growth. Our proof is based on an elegant application of the open mapping theorem, which we complement by a crucial technical lemma allowing us to strengthen our earlier results.

Particles and Spacetime Diagrams

This chapter begins the process of making a 3D interpretation of the plaid model. The idea is to group together certain of the light points and think of them as instances of 1-dimensional worldlines rather than as a succession of points. It fixes an even rational parameter p/q and uses the notation from Section 1.2, i.e., ω‎ = p + q. Section 4.2 explains a different notion of adjacency for the ω‎ × ω‎ blocks, dividing up the plaid model. Section 4.3 says what it means for two horizontal light points in remotely adjacent blocks to be different instances of the same particle. Section 4.4 does the same for the vertical particles. Section 4.5 shows a few pictures of spacetime diagrams and discuss their symmetries. Section 4.6 proves a technical lemma, the Bad Tile Lemma, which is very similar in spirit to Theorem 1.4.

The Core Case

This chapter proves the core case of the Copy Lemma. The proof follows the same strategy as in Chapter 25, but it is considerably harder. Section 26.2 proves the first two statements of the Copy Lemma. The rest of the chapter is devoted to the third statement. Section 26.3 defines geometric and arithmetic alignment as in the previous chapter, but with the twist that there are some extra indices that have to be looked after carefully. Section 26.4 verifies the geometric alignment criterion just as in the previous chapter. Section 26.5 shows that the signs of the two capacity sequences match. Section 26.6 presents a technical lemma to deal with the mass sequences. Section 26.7 shows that the signs of the mass sequences agree on the central indices. Section 26.8 verifies that the signs of the mass sequences agree on the peripheral sequences except for two special indices where the signs can disagree. Section 26.9 finishes the proof of this long and difficult argument.

The Weak and Strong Case

This chapter completes the proof of the weak and strong case of the Copy Lemma. The two cases have just about the same proof. Section 25.2 proves the first two statements of the Copy Lemma. The rest of the chapter is devoted to proving the third statement. Section 25.3 proves an easy technical lemma. Section 25.4 repackages some of the results from Section 1.5. Two sequences are assigned to each rectangle in the plane: a mass sequence and a capacity sequence. It is established that these sequences determine the structure of the plaid model inside the rectangle. Section 25.5 proves a technical result about vertical light points. The final section verifies the conditions of the Matching Criterion.

Properties of the Model

This chapter derives some basic properties of the plaid model. It is organized as follows. Section 2.2 deals with the symmetries of the plaid model. First, it deals with the unoriented model and then considers the oriented model. Section 2.3 proves the technical lemma that each unit integer segment contains exactly two intersection points. The work in this section reveals the nice geometric way that the slanting lines intersect each unit integer square. Section 2.4 establishes the following result: Within each block, there are exactly two lines of capacity k for each even k ɛ [0, ω‎]. Moreover, within the block, each line of capacity k has exactly k light points on it (when double points are appropriately counted). Section 2.5 establishes a subtle additional symmetry of the plaid model.

Independent Sets in the Union of Two Hamiltonian Cycles

Motivated by a question on the maximal number of vertex disjoint Schrijver graphs in the Kneser graph, we investigate the following function, denoted by $f(n,k)$: the maximal number of Hamiltonian cycles on an $n$ element set, such that no two cycles share a common independent set of size more than $k$. We shall mainly be interested in the behavior of $f(n,k)$ when $k$ is a linear function of $n$, namely $k=cn$. We show a threshold phenomenon: there exists a constant $c_t$ such that for $c<c_t$, $f(n,cn)$ is bounded by a constant depending only on $c$ and not on $n$, and for $c_t <c$, $f(n,cn)$ is exponentially large in $n ~(n \to \infty)$. We prove that $0.26 < c_t < 0.36$, but the exact value of $c_t$ is not determined. For the lower bound we prove a technical lemma, which for graphs that are the union of two Hamiltonian cycles establishes a relation between the independence number and the number of $K_4$ subgraphs. A corollary of this lemma is that if a graph $G$ on $n>12$ vertices is the union of two Hamiltonian cycles and $\alpha(G)=n/4$, then $V(G)$ can be covered by vertex-disjoint $K_4$ subgraphs.

A game Pareto complete analysis in n-dimensions: a general applicative study case

The study proposed here considers an applicable game model, in a specific non-linear interfering scenario, with n possible interacting elements. We find the Pareto maximal boundary by using the Carfì’s payoff analysis method for differentiable games. The core section of the paper studies the game by finding the critical zone of the game in its Cartesian form. At this aim, we need to prove an intricate theorem and a technical lemma about the Jacobian determinant of the examined n-game.

Equality of certain automorphism groups of finite p-groups

Let [Formula: see text] be a finite [Formula: see text]-group and let Aut([Formula: see text]) denote the full automorphism group of [Formula: see text]. In the recent past, there has been interest in finding necessary and sufficient conditions on [Formula: see text] such that certain subgroups of Aut([Formula: see text]) are equal. We prove a technical lemma and, as a consequence, obtain some new results and short and alternate proofs of some known results of this type.

Localization at countably infinitely many prime ideals and applications

In this paper we present a technical lemma about localization at countably infinitely many prime ideals. We apply this lemma to get many results about the finiteness of associated prime ideals of local cohomology modules.