Symmetries of degenerate center singularities of plane vector fields

Let $D$ be a closed unit $2$-disk on the plane centered at the origin $O$, and $F$ be a smooth vector field such that $O$ is a unique singular point of $F$ and all other orbits of $F$ are simple closed curves wrapping once around $O$. Thus topologically $O$ is a "center" singularity. Let also $\mathrm{Diff}(F)$ be the group of all diffeomorphisms of $D$ which preserve orientation and orbits of $F$. In arXiv:0907.0359 the author described the homotopy type of $\mathrm{Diff}(F)$ under assumption that the $1$-jet of $F$ at $O$ is non-degenerate. In this paper degenerate case is considered. Under additional "non-degeneracy assumptions" on $F$ the path components of $\mathrm{Diff}(F)$ with respect to distinct weak topologies are described.Comment: 21 page, 3 figure

Homotopy types of stabilizers and orbits of Morse functions on surfaces

Let $M$ be a smooth compact surface, orientable or not, with boundary or without it, $P$ either the real line $R^1$ or the circle $S^1$, and $Diff(M)$ the group of diffeomorphisms of $M$ acting on $C^{\infty}(M,P)$ by the rule $h\cdot f\mapsto f \circ h^{-1}$, where $h\in Diff(M)$ and $f \in C^{\infty}(M,P)$. Let $f:M \to P$ be a Morse function and $O(f)$ be the orbit of $f$ under this action. We prove that $\pi_k O(f)=\pi_k M$ for $k\geq 3$, and $\pi_2 O(f)=0$ except for few cases. In particular, $O(f)$ is aspherical, provided so is $M$. Moreover, $\pi_1 O(f)$ is an extension of a finitely generated free abelian group with a (finite) subgroup of the group of automorphisms of the Reeb graph of $f$. We also give a complete proof of the fact that the orbit $O(f)$ is tame Frechet submanifold of $C^{\infty}(M,P)$ of finite codimension, and that the projection $Diff(M) \to O(f)$ is a principal locally trivial $S(f)$-fibration.Comment: 49 pages, 8 figures. This version includes the proof of the fact that the orbits of a finite codimension of tame action of tame Lie group on tame Frechet manifold is a tame Frechet manifold itsel

Towards Better Separation between Deterministic and Randomized Query Complexity

We show that there exists a Boolean function $F$ which observes the following separations among deterministic query complexity $(D(F))$, randomized zero error query complexity $(R_0(F))$ and randomized one-sided error query complexity $(R_1(F))$: $R_1(F) = \widetilde{O}(\sqrt{D(F)})$ and $R_0(F)=\widetilde{O}(D(F))^{3/4}$. This refutes the conjecture made by Saks and Wigderson that for any Boolean function $f$, $R_0(f)=\Omega({D(f)})^{0.753..}$. This also shows widest separation between $R_1(f)$ and $D(f)$ for any Boolean function. The function $F$ was defined by G{\"{o}}{\"{o}}s, Pitassi and Watson who studied it for showing a separation between deterministic decision tree complexity and unambiguous non-deterministic decision tree complexity. Independently of us, Ambainis et al proved that different variants of the function $F$ certify optimal (quadratic) separation between $D(f)$ and $R_0(f)$, and polynomial separation between $R_0(f)$ and $R_1(f)$. Viewed as separation results, our results are subsumed by those of Ambainis et al. However, while the functions considerd in the work of Ambainis et al are different variants of $F$, we work with the original function $F$ itself.Comment: Reference adde

Restoration of Fricitonal Characteristics on Older Portland Cement Concrete Pavement:Final Report for Iowa Highway Research Board Project HR-224, June 1986

Safety i s a very important aspect o f the highway program. The Iowa DOT initiated an inventory o f the friction values of all paved primary roadways i n 1969. This inventory, with an ASTM E-274 test unit, has continued to the present time. The t e s t i n g frequency varies based upon traffic volume and the previous friction value. Historically , the state o f Iowa constructed a substantial amount o f pcc pavement during the 1928-30 period t o "get Iowa out o f the mud". Some of that pavement has never been resurfaced and has been subjected to more than 50 years o f wear. The textured surface has been worn away and has subsequently polished. Even though some pavements from 15 t o 50 years old continue t o function structurally , because of the loss of friction , they do not provide the desired level o f safety to the driver. As a temporary measure, "Sl ippery -When -Wet " signs have been posted on many older pcc roads due to friction numbers below t h e desirable level. These signs warn the motorist of the current conditions. An economical method of restoring the high quality frictional properties i s needed