On a symplectic generalization of Petrie's conjecture

Motivated by the Petrie conjecture, we consider the following questions: Let a circle act in a Hamiltonian fashion on a compact symplectic manifold $(M,\omega)$ which satisfies H^{2i}(M;\R) = H^{2i}(\CP^n,\R) for all $i$. Is H^j(M;\Z) = H^j(\CP^n;\Z) for all $j$? Is the total Chern class of $M$ determined by the cohomology ring $H^*(M;\Z)$? We answer these questions in the six dimensional case by showing that $H^j(M;\Z)$ is equal to H^j(\CP^3;\Z) for all $j$, by proving that only four cohomology rings can arise, and by computing the total Chern class in each case. We also prove that there are no exotic actions. More precisely, if $H^*(M;\Z)$ is isomorphic to H^*(\CP^3;\Z) or H^*(\Tilde{G}_2(\R^5);\Z), then the representations at the fixed components are compatible with one of the standard actions; in the remaining two case, the representation is strictly determined by the cohomology ring. Finally, our results suggest a natural question: do the remaining two cohomology rings actually arise? This question is closely related to some interesting problems in symplectic topology, such as embeddings of ellipsoids.Comment: 34 pages; accepted to Transactions of the AM

More complete discussion of the time-dependence of the non-static line element for the universe

In a previous article,(1) I have shown that a continuous transformation of matter into radiation, occurring throughout the universe, as postulated by the astrophysicists, would necessitate a nonstatic line element for the universe, and have shown that the non-static character thus introduced might provide an explanation of the red shift in the light from the extra-galactic nebulae. In the present article, I wish to discuss the form of dependence of the line element on the time more completely than was possible on the previous occasion. This is a matter of considerable importance, since changes in the approximations which must be introduced to obtain a usable result affect to quite a different extent the expressions for the relation between red shift and distance and for the rate of annihilation of matter. Indeed, the possibility arises of slight changes from the treatment previously given which would leave the theoretical relation between red shift and distance still approximately linear, as observationally found, and yet produce a very considerable change in the calculated rate for the annihilation of matter

Impact Strength Of Glass And Glass Ceramic

Strength of glass and glass ceramic was measured with a bar impact technique. High-speed movies show regions of tensile and compressive failure. The borosilicatc glass had a compressive strength of at least 2.2 GPa, and the glass ceramic at least 4 GPa. However, the BSG was much stronger in tension than GC. In ballistic tests, the BSG was the superior armor.Mechanical Engineerin

Hamiltonian circle actions on eight dimensional manifolds with minimal fixed sets

Consider a Hamiltonian circle action on a closed $8$-dimensional symplectic manifold $M$ with exactly five fixed points, which is the smallest possible fixed set. In their paper, L. Godinho and S. Sabatini show that if $M$ satisfies an extra "positivity condition" then the isotropy weights at the fixed points of $M$ agree with those of some linear action on $\mathbb{CP}^4$. Therefore, the (equivariant) cohomology rings and the (equivariant) Chern classes of $M$ and $\mathbb{CP}^4$ agree; in particular, $H^*(M;\mathbb{Z}) \simeq \mathbb{Z}[y]/y^5$ and $c(TM) = (1+y)^5$. In this paper, we prove that this positivity condition always holds for these manifolds. This completes the proof of the "symplectic Petrie conjecture" for Hamiltonian circle actions on on 8-dimensional closed symplectic manifolds with minimal fixed sets.Comment: To appear in Transformation Group