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Topology
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
Steve Huntsman, Jimmy Palladino, Michael Robinson
Call ϕ:ℝn→[0,∞)unimodal if ϕ is continuous and the excursion set ϕ−1([y,∞)) is contractible (i.e., homotopy equivalent to a point) for all 0<y≤maxϕ. For n=1, contractibility means that these excursion sets are all intervals, which coincides with the intuitive notion of unimodality. For f:ℝn→[0,∞) sufficiently nice, define the unimodal category of f to be the smallest number M of functions such that f admits a unimodal decomposition of the form f=∑m=1Mπmϕm for some π>0, ∑mπm=1, and ϕm unimodal (Ghrist, 2014).
Introduction
Published in John N. Mordeson, Davender S. Malik, Fuzzy Automata and Languages, 2002
John N. Mordeson, Davender S. Malik
Definition 10.13.8 Let A,B be subspaces of (X,τ). The interior of A, written int A, is defined to be the set, int A={x∈A∣d(x,X∖A)>1}, where d(x,A)=∧x'∈Adx,x'. The boundary of A, written as ∂A, is defined to be the set ∂A=N(A)∖ int A. The tolerance space X is called τ-connected if for all x,x'∈X×X,dx,x'≠∞ and is called contractible if it is homotopy equivalent to a point, i.e., to a tolerance space (Y,δ) with |Y|=1. The component Cx' of x' in X is the maximal τ connected subspace of X containing x'.
Semi-classical solutions for generalized quasilinear Schrödinger equations with nonlocal term and upper critical exponent
Published in Applicable Analysis, 2023
Quanqing Li, Song You, Zuo Yang
We point out that such kind of hypothesis (V) was introduced by Rabinowitz in [35]. Before stating our main results, we introduce some useful notations and definitions. Set with the norm and set with the norm Then (V) implies that the norms and are equivalent. Furthermore, if Y is a given closed set of a topological space X, we denote by the Ljusternik–Schnirelmann category of Y in X, that is the least number of closed and contractible sets in X which cover Y.
Critical point theory for sparse recovery
Published in Optimization, 2023
The formula (26) means that the -skeleton of the -dimensional simplex contains holes of dimension q. By attaching the cells to (22) those q-dimensional holes will be closed. To see this, it is sufficient to show that the union of (22) with the sets from (27) is contractible: It is indeed the case by using the homotopy We conclude that attaching (27) to will result in a contractible set. It remains to note, that is also contractible as shown by the homotopy Since the number of cells in (27) to be attached is , the assertion follows. For the process of cell-attachment see Figure 1, where the special case with p = 3, q = 1 is illustrated.
Existence of positive solutions for a Schrödinger-Poisson system with critical growth
Published in Applicable Analysis, 2020
Next we shall state the existence of multiple positive solutions for (4) by Ljusternik-Schnirelmann category theory. We first recall some basic definitions. If Y is a closed subset of a topological space X, the Ljusternik-Schnirelmann category is the least number of closed and contractible sets in X which cover Y . In order to obtain multiple solutions for , we assume that and let In view of (or ), the set is compact. For any , we define the set . Our main results are as follows.