Borsuk–Ulam theorem for filtered spaces

Let X and Y be pathwise connected and paracompact Hausdorff spaces equipped with free involutions T : X → X {T:X\to X} and S : Y → Y {S:Y\to Y} , respectively. Suppose that there exists a sequence ( X i , T i ) ⁢ ⟶ h i ⁢ ( X i + 1 , T i + 1 ) for ⁢ 1 ≤ i ≤ k , (X_{i},T_{i})\overset{h_{i}}{\longrightarrow}(X_{i+1},T_{i+1})\quad\text{for }% 1\leq i\leq k, where, for each i, X i {X_{i}} is a pathwise connected and paracompact Hausdorff space equipped with a free involution T i {T_{i}} , such that X k + 1 = X {X_{k+1}=X} , and h i : X i → X i + 1 {h_{i}:X_{i}\to X_{i+1}} is an equivariant map, for all 1 ≤ i ≤ k {1\leq i\leq k} . To achieve Borsuk–Ulam-type theorems, in several results that appear in the literature, the involved spaces X in the statements are assumed to be cohomological n-acyclic spaces. In this paper, by considering a more wide class of topological spaces X (which are not necessarily cohomological n-acyclic spaces), we prove that there is no equivariant map f : ( X , T ) → ( Y , S ) {f:(X,T)\to(Y,S)} and we present some interesting examples to illustrate our results.

On the Mazur-Ulam theorem for Fréchet algebras

The Mazur-Ulam theorem for Fréchet algebras is proved.

Fixed-Points of Interpolative Ćirić-Reich–Rus-Type Contractions in b-Metric Spaces

The concept of symmetry is inherent in the study of metric spaces due to the presence of the symmetric property of the metric. Significant results, such as with the Borsuk–Ulam theorem, make use of fixed-point arguments in their proofs to deal with certain symmetry principles. As such, the study of fixed-point results in metric spaces is highly correlated with the symmetry concept. In the current paper, we first define a new and modified Ćirić-Reich–Rus-type contraction in a b-metric space by incorporating the constant s in its definition and establish the corresponding fixed-point result. Next, we adopt an interpolative approach to establish some more fixed-point theorems. Existence of fixed points for ω -interpolative Ćirić-Reich–Rus-type contractions are investigated in this context. We also illustrate the validity of our results with some examples.

PPF-Dependent Fixed Point Results for Multi-Valued ϕ-F-Contractions in Banach Spaces and Applications

The solutions for many real life problems is obtained by interpreting the given problem mathematically in the form of f ( x ) = x . One of such examples is that of the famous Borsuk–Ulam theorem, in which using some fixed point argument, it can be guaranteed that at any given time we can find two diametrically opposite places in a planet with same temperature. Thus, the correlation of symmetry is inherent in the study of fixed point theory. In this paper, we initiate ϕ − F -contractions and study the existence of PPF-dependent fixed points (fixed points for mappings having variant domains and ranges) for these related mappings in the Razumikhin class. Our theorems extend and improve the results of Hammad and De La Sen [Mathematics, 2019, 7, 52]. As applications of our PPF dependent fixed point results, we study the existence of solutions for delay differential equations (DDEs) which have numerous applications in population dynamics, bioscience problems and control engineering.

Generalised Mycielski Graphs and the Borsuk–Ulam Theorem

Stiebitz determined the chromatic number of generalised Mycielski graphs using the topological method of Lovász, which invokes the Borsuk–Ulam theorem. Van Ngoc and Tuza used elementary combinatorial arguments to prove Stiebitz's theorem for 4-chromatic generalised Mycielski graphs, and asked if there is also an elementary combinatorial proof for higher chromatic number. We answer their question by showing that Stiebitz's theorem can be deduced from a version of Fan's combinatorial lemma. Our proof uses topological terminology, but is otherwise completely discrete and could be rewritten to avoid topology altogether. However, doing so would be somewhat artificial, because we also show that Stiebitz's theorem is equivalent to the Borsuk–Ulam theorem.

On the Chromatic Number of Matching Kneser Graphs

In an earlier paper, the present authors (2015) introduced the altermatic number of graphs and used Tucker’s lemma, an equivalent combinatorial version of the Borsuk–Ulam theorem, to prove that the altermatic number is a lower bound for chromatic number. A matching Kneser graph is a graph whose vertex set consists of all matchings of a specified size in a host graph and two vertices are adjacent if their corresponding matchings are edge-disjoint. Some well-known families of graphs such as Kneser graphs, Schrijver graphs and permutation graphs can be represented by matching Kneser graphs. In this paper, unifying and generalizing some earlier works by Lovász (1978) and Schrijver (1978), we determine the chromatic number of a large family of matching Kneser graphs by specifying their altermatic number. In particular, we determine the chromatic number of these matching Kneser graphs in terms of the generalized Turán number of matchings.