In this note, we show that cite[Corollary 3.2]{sad} is not always true. In fact, we characterize essential left $phi$-contractibility of the group algebras in terms of compactness of its related locally compact group. Also, we show that for any compact commutative group $G$, $L^{2}(G)$ is always essentially left $phi$-contractible. We discuss the essential left $phi$-contractibility of some Fou...

We prove that a triangle-free graph drawn in the torus with all faces bounded by even walks is 3-colorable if and only if it has no subgraph isomorphic to the Cayley graph C(Z13;1,5). We also prove that a non-bipartite quadrangulation of the Klein bottle is 3colorable if and only if it has no non-contractible separating cycle of length at most four and no odd walk homotopic to a non-contractibl...

We prove the following three results in Hamiltonian dynamics. • The Weinstein conjecture holds true for every displaceable hypersurface of contact type. • Every magnetic flow on a closed Riemannian manifold has contractible closed orbits for a dense set of small energies. • Every closed Lagrangian submanifold whose fundamental group injects and which admits a Riemannian metric without closed ge...

An element e of a 3-connected matroid M is said to be contractible provided that M/e is 3-connected. In this paper, we show that a 3-connected matroid M with exactly k contractible elements has at least max { r(M) + 6− 2k 4 , |E(M)| + 6− 3k 5 } triangles. For each k, we construct an infinite family of matroids that attain this bound. New sharp bounds for the number of triads of a minimally 3-co...

Here Dfij = ∂fi/∂zj is the Jacobian matrix of f , J is the Poisson matrix, and I is the n × n identity matrix. Equivalently, Stokes’ theorem can be used to show that the loop action, A[γ] = ∮ γ pdq, is preserved by f for any contractible loop γ on X. If f preserves the loop action for all loops, even those that are not contractible, then it is exact symplectic. When n = 1, the symplectic condit...

A structural description is given of the separable C∗-algebras that are contractible in KK-theory or E-theory. Subsequently it is shown that two separable C∗-algebras, A and B, are KK-equivalent if and only if there is a bundle of separable C∗-algebras over [0, 1] which is piecewise trivial with no more than six points of non-triviality such that the kernel of each fiber map is KK-contractible,...

In Stanley’s seminal 1995 paper on the chromatic symmetric function, he stated that there was no known graph that was not contractible to the claw and whose chromatic symmetric function was not e-positive, namely, not a positive linear combination of elementary symmetric functions. We resolve this by giving infinite families of graphs that are not contractible to the claw and whose chromatic sy...