Under the assumption of proportional hazards, the log-rank test is optimal for testing the null hypothesis of H0 : β = 0, where β denotes the logarithm of the hazard ratio. However, if there are additional covariates that correlate with survival times, making use of their information will increase the efficiency of the log-rank test. We apply the theory of semiparametrics to characterize a clas...

It was only recently shown by Shi and Wormald, using the differential equation method to analyse an appropriate algorithm, that a random 5-regular graph asymptotically almost surely has chromatic number at most 4. Here, we show that the chromatic number of a random 5-regular graph is asymptotically almost surely equal to 3, provided a certain four-variable function has a unique maximum at a giv...

We show that on every Ramanujan graph G, the simple random walk exhibits cutoff: when G has n vertices and degree d, the total-variation distance of the walk from the uniform distribution at time t = d d−2 logd−1 n + s √ logn is asymptotically P(Z > c s) where Z is a standard normal variable and c = c(d) is an explicit constant. Furthermore, for all 1 ≤ p ≤ ∞, d-regular Ramanujan graphs minimiz...

An upper bound is given on the minimum distance between i subsets of the same size of a regular graph in terms of the i-th largest eigenvalue in absolute value. This yields a bound on the diameter in terms of the i-th largest eigenvalue, for any integer i. Our bounds are shown to be asymptotically tight. A recent result by Quenell relating the diameter, the second eigenvalue, and the girth of a...

Let \((X,\mathscr {B}, \mu ,T,d)\) be a measure-preserving dynamical system with exponentially mixing property, and let \(\mu \) an Ahlfors s-regular probability measure. The covering problem concerns the set E(x) of points which are covered by orbits \(x\in X\) infinitely many times. We prove that Hausdorff dimension intersection any regular fractal G \(\dim _\mathrm{H}G>s-\alpha equals _\math...

We construct a new class of asymptotically flat black hole solutions in EinsteinYang-Mills and Einstein-Yang-Mills-dilaton theory. These black hole solutions are static, and they have a regular event horizon. However, they possess only axial symmetry. Like their regular counterparts, the black hole solutions are characterized by two integers, the winding number n and the node number k of the ga...

We give a construction of quantum LDPC codes dimension $\Theta(\log N)$ and distance $\Theta(N/\log as the code length $N\to\infty$. Using product chain complexes this also provides family $\Omega(N^{1-\alpha/2}/\log $\Omega(N^\alpha \log N)$, where $0 \le \alpha < 1$. introduce study new operation called lifted product, which naturally generalizes operations for complexes. Moreover, simple byp...

Asymptotic properties of random regular graphs are object of extensive study in mathematics. In this note we argue, based on theory of spin glasses, that in random regular graphs the maximum cut size asymptotically equals the number of edges in the graph minus the minimum bisection size. Maximum cut and minimal bisection are two famous NP-complete problems with no known general relation between...

We consider the dynamics of even solutions one-dimensional nonlinear Klein-Gordon equation $\partial_t^2 \phi - \partial_x^2 + |\phi|^{2\alpha} =0$ for $\alpha>1$, in vicinity unstable soliton $Q$. Our main result is that stability energy space $H^1(\mathbb R)\times L^2(\mathbb R)$ implies asymptotic a local norm. In particular, there exists Lipschitz graph initial data leading to stable and as...