1 The Borsuk-Ulam theorem We have seen how combinatorics borrows from probability theory. Another area which has been very beneficial to combinatorics, perhaps even more surprisingly, is topology. We have already seen Brouwer's fixed point theorem and its combinatorial proof. Theorem 1 (Brouwer). For any continuous function f : B n → B n , there is a point x ∈ B n such that f (x) = x. A more po...

The existence of non-trivial solutions X to matrix equations of the form F (X,A1,A2, · · · ,As) = G(X,A1,A2, · · · ,As) over the real numbers is investigated. Here F and G denote monomials in the (n × n)-matrix X = (xij) of variables together with (n × n)-matrices A1,A2, · · · ,As for s ≥ 1 and n ≥ 2 such that F and G have different total positive degrees in X. An example with s = 1 is given by...

Part II of this study uses the path-following theory of labelled V-complexes developed in Part I to provide constructive algorithmic proofs of a variety of combinatorial lenmas in topology. \Je demonstrate two new dual lemmas on the n-dimensional cube, and use a Generc.lized S ^erner Lemma to prove a gener lization of the Knaster-Kuratowski-y.azurkiewicz Covering Lemma on the simplex. We also s...

This theorem has a number of geometric and number theoretic consequences that will be discussed in the final section of this paper. In particular, assuming Lang’s conjecture on rational points of varieties of general type, we can prove a uniform bound on the number of rational points on a surface of general type not contained in rational or elliptic curves. This theorem is a special case of the...

The present review paper provides recent results on convexity and its applications to the constrained extension of linear operators, motivated by existence subgradients continuous convex Markov moment problem related approximation using Krein–Milman theorem, optimization, polynomial unbounded subsets. In many cases, Mazur–Orlicz theorem also leads operators as solutions. common point all these ...

In this paper, we obtain the Hyers–Ulam–Rassias stability of the generalized Pexider functional equation ∑ k∈K f(x+ k · y) = |K|g(x) + |K|h(y), x, y ∈ G, where G is an abelian group, K is a finite abelian subgroup of the group of automorphism of G. The concept of Hyers–Ulam–Rassias stability originated from Th.M. Rassias’ Stability Theorem that appeared in his paper: On the stability of the lin...

(Joint work with Alexandru Chirvasitu.) The Borsuk-Ulam theorem in algebraic topology indicates restrictions for equivariant maps between spheres; in particular, there is no odd map from a sphere to another sphere of lower dimension. This idea may be generalized greatly in both the topological and operator algebraic settings for actions of compact (quantum) groups, leading to the the noncommuta...

‎In this paper‎, ‎using fixed point method‎, ‎we prove the generalized Hyers-Ulam stability of‎ ‎random homomorphisms in random $C^*$-algebras and random Lie $C^*$-algebras‎ ‎and of derivations on Non-Archimedean random C$^*$-algebras and Non-Archimedean random Lie C$^*$-algebras for‎ ‎the following $m$-variable additive functional equation:‎ ‎$$sum_{i=1}^m f(x_i)=frac{1}{2m}left[sum_{i=1}^mfle...

If the answer is affirmative, the functional equation for homomorphisms is said to be stable in the sense of Hyers and Ulam because the first result concerning the stability of functional equations was presented by Hyers. Indeed, he has answered the question of Ulam for the case where G1 and G2 are assumed to be Banach spaces (see [8]). We may find a number of papers concerning the stability re...