Counting Labelled Trees with Given Indegree Sequence

For a labelled tree on the vertex set $[n]:=\{1,2,..., n\}$, define the direction of each edge $ij$ to be $i\to j$ if $i<j$. The indegree sequence of $T$ can be considered as a partition $\lambda \vdash n-1$. The enumeration of trees with a given indegree sequence arises in counting secant planes of curves in projective spaces. Recently Ethan Cotterill conjectured a formula for the number of trees on $[n]$ with indegree sequence corresponding to a partition $\lambda$. In this paper we give two proofs of Cotterill's conjecture: one is `semi-combinatorial" based on induction, the other is a bijective proof.Comment: 10 page

Performance of Cross-layer Design with Multiple Outdated Estimates in Multiuser MIMO System

By combining adaptive modulation (AM) and automatic repeat request (ARQ) protocol as well as user scheduling, the cross-layer design scheme of multiuser MIMO system with imperfect feedback is presented, and multiple outdated estimates method is proposed to improve the system performance. Based on this method and imperfect feedback information, the closed-form expressions of spectral efficiency (SE) and packet error rate (PER) of the system subject to the target PER constraint are respectively derived. With these expressions, the system performance can be effectively evaluated. To mitigate the effect of delayed feedback, the variable thresholds (VTs) are also derived by means of the maximum a posteriori method, and these VTs include the conventional fixed thresholds (FTs) as special cases. Simulation results show that the theoretical SE and PER are in good agreement with the corresponding simulation. The proposed CLD scheme with multiple estimates can obtain higher SE than the existing CLD scheme with single estimate, especially for large delay. Moreover, the CLD scheme with VTs outperforms that with conventional FTs

Spin relaxation and decoherence of two-level systems

We revisit the concepts of spin relaxation and spin decoherence of two level (spin-1/2) systems. From two toy-models, we clarify two issues related to the spin relaxation and decoherence: 1) For an ensemble of two-level particles each subjected to a different environmental field, there exists an ensemble relaxation time $T_1^*$ which is fundamentally different from $T_1$. When the off-diagonal coupling of each particle is in a single mode with the same frequency but a random coupling strength, we show that $T_1^*$ is finite while the spin relaxation time of a single spin $T_1$ and the usual ensemble decoherence time $T_2^*$ are infinite. 2) For a two-level particle under only a random diagonal coupling, its relaxation time $T_1$ shall be infinite but its decoherence time $T_2$ is finite.Comment: 5 pages, 2 figure