Packing Directed Cycles Quarter- and Half-Integrally

The celebrated Erd\H{o}s-P\'osa theorem states that every undirected graph that does not admit a family of $k$ vertex-disjoint cycles contains a feedback vertex set (a set of vertices hitting all cycles in the graph) of size $O(k \log k)$. After being known for long as Younger's conjecture, a similar statement for directed graphs has been proven in 1996 by Reed, Robertson, Seymour, and Thomas. However, in their proof, the dependency of the size of the feedback vertex set on the size of vertex-disjoint cycle packing is not elementary. We show that if we compare the size of a minimum feedback vertex set in a directed graph with the quarter-integral cycle packing number, we obtain a polynomial bound. More precisely, we show that if in a directed graph $G$ there is no family of $k$ cycles such that every vertex of $G$ is in at most four of the cycles, then there exists a feedback vertex set in $G$ of size $O(k^4)$. Furthermore, a variant of our proof shows that if in a directed graph $G$ there is no family of $k$ cycles such that every vertex of $G$ is in at most two of the cycles, then there exists a feedback vertex set in $G$ of size $O(k^6)$. On the way there we prove a more general result about quarter-integral packing of subgraphs of high directed treewidth: for every pair of positive integers $a$ and $b$, if a directed graph $G$ has directed treewidth $\Omega(a^6 b^8 \log^2(ab))$, then one can find in $G$ a family of $a$ subgraphs, each of directed treewidth at least $b$, such that every vertex of $G$ is in at most four subgraphs.Comment: Accepted to European Symposium on Algorithms (ESA '19

Half-integral Erd\H{o}s-P\'osa property of directed odd cycles

We prove that there exists a function $f:\mathbb{N}\rightarrow \mathbb{R}$ such that every digraph $G$ contains either $k$ directed odd cycles where every vertex of $G$ is contained in at most two of them, or a vertex set $X$ of size at most $f(k)$ hitting all directed odd cycles. This extends the half-integral Erd\H{o}s-P\'osa property of undirected odd cycles, proved by Reed [Mangoes and blueberries. Combinatorica 1999], to digraphs.Comment: 16 pages, 5 figure

Lithium/sulfur dioxide cell and battery safety

The new high-energy lithium/sulfur dioxide primary electrochemical cell, having a number of advantages, has received considerable attention as a power source in the past few years. With greater experience and improved design by the manufacturers, this system can be used in a safe manner provided the guidelines for use and safety precautions described herein are followed. In addition to a description of cell design and appropriate definitions, there is a safety precautions checklist provided to guide the user. Specific safety procedures for marking, handling, transportation, and disposal are also given, as is a suggested series of tests, to assure manufacturer conformance to requirements

The 110 watt per kilogram lightweight solar array system

This ultralightweight solar array system is applied to three generic mission types: (1) interplanetary, (2) geosynchronous, and (3) manned space station. The requirements of each of these missions, as they pertain to the solar array, are presented. Existing lightweight solar array system concepts are reviewed, along with conclusions regarding the applicability of this technology to the feasibility of the ultralightweight solar array system. Several system concepts are included for further evaluation. The existing technology base, as it pertains to solar cells, solar cell covers, interconnects and substrates, and deployable booms, is reviewed. The attitude control of spacecraft with large flexible solar arrays is also discussed

Space programs summary no. 37-49, volume 3 for the period December 1, 1967 to January 30, 1968. Supporting research and advanced development

Space program research projects on systems analysis and engineering, telecommunications, guidance and control, propulsion, and data system

Liquid Rocket Booster (LRB) for the Space Transportation System (STS) systems study, volume 2

The Liquid Rocket Booster (LRB) Systems Definition Handbook presents the analyses and design data developed during the study. The Systems Definition Handbook (SDH) contains three major parts: the LRB vehicles definition; the Pressure-Fed Booster Test Bed (PFBTB) study results; and the ALS/LRB study results. Included in this volume are the results of all trade studies; final configurations with supporting rationale and analyses; technology assessments; long lead requirements for facilities, materials, components, and subsystems; operational requirements and scenarios; and safety, reliability, and environmental analyses

Hard X-ray imaging facility for space shuttle: A scientific and conceptual engineering study

A shuttle-accommodated instrument for imaging hard X-rays in the study of nonthermal particles and high temperature particles in various solar and cosmic phenomena was defined and its feasibility demonstrated. The imaging system configuration is described as well as the electronics, aspect systems, mechanical and thermal properties and the ground support equipment