It is only recently that a renewed special attention has been drawn to the study of contagion dynamics over higher-order interactions and over more general graph structures, like simplexes. Classically, the multi-group SIS model has assumed pairwise interactions of contagion across groups and thus has been vastly studied in the literature. We focus on simplicial complexes and refer to the model as to the simplicial SIS model. This paper analyzes a Susceptible-Infected-Susceptible (SIS) model of epidemic propagation over hypergraphs. 2) makes another appearance, this time to prove a far-reaching generalization of the Schröder–Bernstein Theorem. For this result the fixed-point theorem of Knaster and Tarski (Theorem 1. We observed that existence of such a “mean” is equivalent to existence of a finitely additive probability “measure” on \(\mathcal\) with nonvoid interior, each can be partitioned into a finite collection of subsets that can be rigidly reassembled into the other. 10 we used the fixed-point theorem of Markov and Kakutani to show that every abelian group G is “amenable” in the sense that there is a G-invariant mean on the vector space B(G) of bounded, real-valued functions on G. This result, which far surpassed anything that seemed attainable at the time, is only part of what Lomonosov proved in an astonishing two-page paper that introduced nonlinear methods-in particular the Schauder Fixed-Point Theorem-into this supposedly hard-core-linear area of mathematics. If an operator T on a Banach space commutes with a non-zero compact operator, then T has a nontrivial invariant subspace. In this chapter we’ll see why invariant subspaces are of interest and then will prove one of the subject’s landmark theorems: Victor Lomonosov’s 1973 result, a special case of which states: For Hilbert space, however, the Invariant Subspace Problem remains open, and is the subject of much research. Here “operator” means “continuous linear transformation,” and “invariant subspace” means “closed (linear) subspace that the operator takes into itself.” To say that a subspace is “nontrivial” means that it is neither the zero subspace nor the whole space.Įxamples constructed toward the end of the last century show that in the generality of Banach spaces there do exist operators with only trivial invariant subspaces. Does every operator on Hilbert space have a nontrivial invariant subspace? This chapter is about the most vexing problem in the theory of linear operators on Hilbert space: The Schauder Theorem will also be important in the next chapter where it will provide a key step in the proof of Lomonosov’s famous theorem on invariant subspaces for linear operators on Banach spaces. After proving the theorem we’ll use it to prove an important generalization of the Picard–Lindelöf Theorem of Chap. This is the famous Schauder Fixed-Point Theorem (circa 1930) which will occupy us throughout this chapter. However all is not lost: the “convex” version does survive: compact, convex subsets of normed linear space do have the fixed-point property. It turns out that the “ball” version of Brouwer’s theorem does not survive the transition to infinitely many dimensions. The closed unit ball ofĪnd the seemingly more general, but in fact equivalent 4 we proved two versions of the Brouwer Fixed-Point Theorem: The “Ball” version (Theorem 4. Recall that to say a metric space has the fixed-point property means that every continuous mapping taking the space into itself must have a fixed point. The material is split into four parts: the first introduces the Banach Contraction-Mapping Principle and the Brouwer Fixed-Point Theorem, along with a selection of interesting applications the second focuses on Brouwer’s theorem and its application to John Nash’s work the third applies Brouwer’s theorem to spaces of infinite dimension and the fourth rests on the work of Markov, Kakutani, and Ryll–Nardzewski surrounding fixed points for families of affine maps. Readers will find the presentation especially useful for independent study or as a supplement to a graduate course in fixed-point theory. Appendices provide an introduction to (or refresher on) some of the prerequisite material and exercises are integrated into the text, contributing to the volume’s ability to be used as a self-contained text. The level of exposition increases gradually throughout the book, building from a basic requirement of undergraduate proficiency to graduate-level sophistication. This text provides an introduction to some of the best-known fixed-point theorems, with an emphasis on their interactions with topics in analysis.
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