We describe an explicit chain map from the standard resolution to the minimal resolution for the finite cyclic group Zk of order k. We then demonstrate how such a chain map induces a “Zk-combinatorial Stokes theorem”, which in turn implies “Dold’s theorem” that there is no equivariant map from an n-connected to an n-dimensional free Zk-complex. Thus we build a combinatorial access road to probl...

We prove a local version of the Mazur-Ulam theorem.

Extensions of the Borsuk-Ulam Theorem by Timothy Prescott

The pioneering Fermi–Pasta–Ulam (FPU) numerical experiment played a major role in the history of computer simulation because it introduced this concept for the first time. Moreover, it raised a puzzling question which was answered more than 10 years later. After an introduction to this problem, we briefly review its history and then suggest some simple numerical experiments, with the Matlab© co...

In 1940, Ulam proposed the stability problem see 1 : Let G1 be a group and let G2 be a metric group with the metric d ·, · . Given ε > 0, does there exist a δ > 0 such that if a mapping h : G1 → G2 satisfies the inequality d h xy , h x h y < δ for all x, y ∈ G1 then there is a homomorphism H : G1 → G2 with d h x ,H x < ε for all x ∈ G1? In 1941, this problem was solved by Hyers 2 in the case of...

In 1940, Ulam proposed the general Ulam stability problem see 1 . Let G1 be a group and let G2 be a metric group with the metric d ·, · . Given ε > 0, does there exist a δ > 0 such that if a mapping h : G1 → G2 satisfies the inequality d h xy , h x h y < δ for all x, y ∈ G1 then there is a homomorphism H : G1 → G2 with d h x ,H x < ε for all x ∈ G1? In 1941, this problem was solved by Hyers 2 i...

Aoki,T. , (1950) &quot;On stability of the linear transformation in Banach spaces,&quot; Journal of the Mathematical Society of Japan, 2,64-66 Chang, S. and Kim,H. M. , (2002), On the Hyer-Ulam stability of a quadratic functional equations, J. Ineq. Appl. Math. , 33, 1-12. Chang,S. , Lee,E. H. and Kim,H. M. , (2003)On the Hyer-Ulam Rassias stability of a quadratic functional equations, Math. In...

In this paper we study the Ulam-Hyers stability of some linear and nonlinear dynamic equations and integral equations on time scales. We use both direct and operatorial methods and we propose a unified approach to Ulam-Hyers stability based on the theory of Picard operators (see [29] and[34]). Our results extend some recent results from [25],[26], [8], [14], [13] to dynamic equations and are mo...

Aoki,T. , (1950) &quot;On stability of the linear transformation in Banach spaces,&quot; Journal of the Mathematical Society of Japan, 2,64-66 Chang, S. and Kim,H. M. , (2002), On the Hyer-Ulam stability of a quadratic functional equations, J. Ineq. Appl. Math. , 33, 1-12. Chang,S. , Lee,E. H. and Kim,H. M. , (2003)On the Hyer-Ulam Rassias stability of a quadratic functional equations, Math. In...

Aoki,T. , (1950) &quot;On stability of the linear transformation in Banach spaces,&quot; Journal of the Mathematical Society of Japan, 2,64-66 Chang, S. and Kim,H. M. , (2002), On the Hyer-Ulam stability of a quadratic functional equations, J. Ineq. Appl. Math. , 33, 1-12. Chang,S. , Lee,E. H. and Kim,H. M. , (2003)On the Hyer-Ulam Rassias stability of a quadratic functional equations, Math. In...