Time-Dependent Vacuum Energy Induced by D-Particle Recoil

We consider cosmology in the framework of a `material reference system' of D particles, including the effects of quantum recoil induced by closed-string probe particles. We find a time-dependent contribution to the cosmological vacuum energy, which relaxes to zero as $\sim 1/ t^2$ for large times $t$. If this energy density is dominant, the Universe expands with a scale factor $R(t) \sim t^2$. We show that this possibility is compatible with recent observational constraints from high-redshift supernovae, and may also respect other phenomenological bounds on time variation in the vacuum energy imposed by early cosmology.Comment: 14 pages LATEX, no figure

An improvement of sufficient condition for kk-leaf-connected graphs

For integer $k\geq2,$ a graph $G$ is called $k$-leaf-connected if $|V(G)|\geq k+1$ and given any subset $S\subseteq V(G)$ with $|S|=k,$ $G$ always has a spanning tree $T$ such that $S$ is precisely the set of leaves of $T.$ Thus a graph is $2$-leaf-connected if and only if it is Hamilton-connected. In this paper, we present a best possible condition based upon the size to guarantee a graph to be $k$-leaf-connected, which not only improves the results of Gurgel and Wakabayashi [On $k$-leaf-connected graphs, J. Combin. Theory Ser. B 41 (1986) 1-16] and Ao, Liu, Yuan and Li [Improved sufficient conditions for $k$-leaf-connected graphs, Discrete Appl. Math. 314 (2022) 17-30], but also extends the result of Xu, Zhai and Wang [An improvement of spectral conditions for Hamilton-connected graphs, Linear Multilinear Algebra, 2021]. Our key approach is showing that an $(n+k-1)$-closed non-$k$-leaf-connected graph must contain a large clique if its size is large enough. As applications, sufficient conditions for a graph to be $k$-leaf-connected in terms of the (signless Laplacian) spectral radius of $G$ or its complement are also presented.Comment: 15 pages, 2 figure

Calculating Ramsey Numbers by Partitioning Colored Graphs

In this paper we prove a new result about partitioning coloured complete graphs and use it to determine certain Ramsey Numbers exactly. The partitioning theorem we prove is that for k ≥ 1, in every edge colouring of Kn with the colours red and blue, it is possible to cover all the vertices with k disjoint red paths and a disjoint blue balanced complete (k+1)-partite graph. When the colouring of Kn is connected in red, we prove a stronger result—that it is possible to cover all the vertices with k red paths and a blue balanced complete (k + 2)-partite graph. Using these results we determine the Ramsey Number of a path, Pn, versus a balanced complete t-partite graph on tm vertices, Kt m, whenever m ≡ 1 (mod n−1). We show that in this case R(Pn, Kt m) = (t − 1)(n − 1) + t(m − 1) + 1, generalizing a result of Erd˝os who proved the m = 1 case of this result. We also determine the Ramsey Number of a path Pn versus the power of a path P t n . We show that R(Pn, Pt n ) = t(n − 1) + j n t+1 k , solving a conjecture of Allen, Brightwell, and Skokan

On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields

Let $v$ be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform \operatorname H_{v, \epsilon}f(x) := \text{p.v.}\int_{-\epsilon}^ \epsilon f(x-yv(x)) \frac{dy}y where $\epsilon$ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.\thinspace M.\thinspace Stein, that if $v$ is Lipschitz, there is a positive $\epsilon$ for which the transform above is bounded on $L ^{2}$. Our principal result gives a sufficient condition in terms of the boundedness of a maximal function associated to $v$. This sufficient condition is that this new maximal function be bounded on some $L ^{p}$, for some $1<p<2$. We show that the maximal function is bounded from $L ^{2}$ to weak $L ^{2}$ for all Lipschitz maximal function. The relationship between our results and other known sufficient conditions is explored.Comment: 92 pages, 20+ figures. Final version of the paper. To appear in Memoirs AM