A property on reinforcing edge-disjoint spanning hypertrees in uniform hypergraphs

Suppose that $H$ is a simple uniform hypergraph satisfying $\left|E\left(H\right)\right|=k(\left|V\left(H\right)\right|?1)$. A $k$-partition $\pi =(X_1,X_2,\dots ,X_k)$ of $E\left(H\right)$ such that $\left|X_i\right|=\left|V\left(H\right)\right|?1$ for $1\le i\le k$ is a uniform $k$-partition. Let $P_k\left(H\right)$ be the collection of all uniform $k$-partitions of $E\left(H\right)$ and define $\epsilon \left(\pi \right)={\sum }_{i=1}^{k}c\left(H\left(X_i\right)\right)?k$, where $c\left(H\right)$ denotes the number of maximal partition-connected sub-hypergraphs of $H$. Let $\epsilon \left(H\right)={min}_{\pi \in P_k\left(H\right)}\epsilon \left(\pi \right)$. Then $\epsilon \left(H\right)\ge 0$ with equality holds if and only if $H$ is a union of $k$ edge-disjoint spanning hypertrees. The parameter $\epsilon \left(H\right)$ is used to measure how close $H$ is being from a union of $k$ edge-disjoint spanning hypertrees.We prove that if $H$ is a simple uniform hypergraph with $\left|E\left(H\right)\right|=k(\left|V\left(H\right)\right|?1)$ and $\epsilon \left(H\right)>0$, then there exist $e\in E\left(H\right)$ and $e^{\prime }\in E\left(H^c\right)$ such that $\epsilon (H?e+e^{\prime })<\epsilon \left(H\right)$. This generalizes a former result, which settles a conjecture of Payan. The result iteratively defines a finite $\epsilon$-decreasing sequence of uniform hypergraphs $H_0,H_1,H_2,\dots ,H_m$ such that $H_0=H$, $H_m$ is the union of $k$ edge-disjoint spanning hypertrees, and such that two consecutive hypergraphs in the sequence differ by exactly one hyperedge.     

Asymptotic models for a long,elastic cylinder

We study asymptotic expansions for the displacement field of a long elastic cylinder under various constitutive assumptions. We show that under simple hypotheses it is possible to derive from the equations of continuum mechanics two known beam equations and several different string models. Some of the string models correspond to those studied by S. Antman and R. Dickey. We also show that under our assumptions the problem of asymptotic expansion can be reduced to that of algebraic geometry.Research partially supported by NSF Grant DMR-8612369.     

Connectivity keeping stars or double-stars in 2-connected graphs

In Mader (2010), Mader conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with $\delta \left(G\right)\ge ?\frac{3}{2}k?+m?1$ contains a subtree $T^{\prime }$ isomorphic to $T$ such that $G?V\left(T^{\prime }\right)$ is $k$-connected. In the same paper, Mader proved that the conjecture is true when $T$ is a path. Diwan and Tholiya (2009) verified the conjecture when $k=1$. In this paper, we will prove that Mader’s conjecture is true when $T$ is a star or double-star and $k=2$.     

On the Semi-geostrophic System in Physical Space with General Initial Data

Archive for Rational Mechanics and Analysis - In order to accommodate general initial data, an appropriately relaxed notion of renormalized Lagrangian solutions for the Semi-Geostrophic system in...     

Cleaning Random d-Regular Graphs with Brooms

A model for cleaning a graph with brushes was recently introduced. Let α = (v ₁, v ₂, . . . , v _n ) be a permutation of the vertices of G; for each vertex v _i let ${N^+(v_i)=\{j: v_j v_i \in E {\rm and} j>\,i\}}${N^+(v_i)=\{j: v_j v_i \in E {\rm and} j>\,i\}} and N^-(v_i)={j: v_j v_i ? E and j <  i}{N^-(v_i)=\{j: v_j v_i \in E {\rm and} j<\,i\}} ; finally let b_a(G)=?_i=1ⁿ max{|N⁺(v_i)|-|N^-(v_i)|,0}{b_{\alpha}(G)=\sum_{i=1}^n {\rm max}\{|N^+(v_i)|-|N^-(v_i)|,0\}}. The Broom number is given by B(G) =  max _α b _α (G). We consider the Broom number of d-regular graphs, focusing on the asymptotic number for random d-regular graphs. Various lower and upper bounds are proposed. To get an asymptotically almost sure lower bound we use a degree-greedy algorithm to clean a random d-regular graph on n vertices (with dn even) and analyze it using the differential equations method (for fixed d). We further show that for any d-regular graph on n vertices there is a cleaning sequence such at least n(d + 1)/4 brushes are needed to clean a graph using this sequence. For an asymptotically almost sure upper bound, the pairing model is used to show that at most n(d+2?{d ln2})/4{n(d+2\sqrt{d \ln 2})/4} brushes can be used when a random d-regular graph is cleaned. This implies that for fixed large d, the Broom number of a random d-regular graph on n vertices is asymptotically almost surely \fracn4(d+Q(?d)){\frac{n}{4}(d+\Theta(\sqrt{d}))}.