Given a poset P, a family F of elements in the Boolean lattice is said to be P-saturated if (1) F contains no copy of P as a subposet and (2) every proper superset of F contains a copy of P as a subposet. The maximum size of a P-saturated family is denoted by La(n, P), which has been studied for a number of choices of P. The minimum size of a P-saturated family, sat(n, P), was introduced by Gerbner et al. (2013), and parallels the deep literature on the saturation function for graphs.We introduce and study the concept of saturation for induced subposets. As opposed to induced saturation in graphs, the above definition of saturation for posets extends naturally to the induced setting. We give several exact results and a number of bounds on the induced saturation number for several small posets. We also use a transformation to the biclique cover problem to prove a logarithmic lower bound for a rich infinite family of target posets. †

No abstract

No abstract

The strong chromatic index of a graph G, denoted χ ′ s (G), is the least number of colors needed to edge-color G so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted χ ′ ℓ,s (G), is the least integer k such that if arbitrary lists of size k are assigned to each edge then G can be edge-colored from those lists where edges at distance at most two receive distinct colors. We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if G is a subcubic planar graph with girth(G) ≥ 41 then χ ′ ℓ,s (G) ≤ 5, answering a question of Borodin and Ivanova [Precise upper bound for the strong edge chromatic number of sparse planar graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759-770]. We further show that if G is a subcubic planar graph and girth(G) ≥ 30, then χ ′ s (G) ≤ 5, improving a bound from the same paper. Finally, if G is a planar graph with maximum degree at most four and girth(G) ≥ 28, then χ ′ s (G) ≤ 7, improving a more general bound of Wang and Zhao from [Odd graphs and its application on the strong edge coloring, arXiv:1412.8358] in this case.

Drill bits are iteratively developed for specific applications to meet performance objectives such as aggressiveness, durability, stability, steerability, etc. The transition from one iteration to the next occurs when dull bits are examined, run data is analyzed and the inferences are implemented as revisions to the bit design and/or the operating parameters. Experience has shown that the efficiency of the process depends strongly upon the appropriateness and significance of the data collected. If, for example, a dominant cutter failure mechanism is incorrectly characterized as abrasive wear and is, in fact, impact wear, then the proposed solution and the development time will both be significantly affected. One method for improving the significance of the data collected is to implement a special-purpose data-acquisition system within the bit. Such a device has at least three significant advantages over available sub-based data acquisition systems:its bit-based sensors will more accurately detect bit-based events,it can be economically deployed over many bits andit is relatively transparent to the user. The paper describes an effort to develop, validate, and utilize a bit-based module designed to monitor accelerometer and magnetometer sensors and to record selected data. The authors also describe the effort to infer dynamic dysfunction(s) from the data in the fast-growing hard rock PDC bit application and to mitigate them via modifications to drilling parameters and bit designs. Background Rotary rock bits are developed iteratively. Typically, the process begins when a performance deficiency is observed in the field. During an investigation phase, dull bits and associated run data are examined under the performance objective. Performance objectives are usually tailored for particular applications and stated in terms such as aggressiveness, durability, stability, steerability, etc. During a proposal phase, a hypothesis is formed in an attempt to explain the deficiency in terms of bit design and / or operating parameters and a solution is proposed. During the test phase, the proposed solution is incorporated into either test bits or into operating practices and then evaluated. If the proposed solution is justified, the process ends and insight is gained; otherwise and more commonly, the next iteration is started. Experience has shown that the efficiency of the development process is strongly dependent on the appropriateness and significance of the data collected during the investigation phase. If, for example, a dominant cutter failure mechanism is incorrectly characterized as abrasive wear and is, in fact, impact wear, then the proposed solution and the development time will both be significantly affected. Representative nearbit dynamic data are needed to directly support drill bit and drilling parameter development programs. Indirectly, the data are also needed to validate laboratory simulation tests and software. There is a long history of researchers trying to understand down-hole dynamic dysfunctions. In the 1960s, Esso Research reported accelerations measured down-hole. Since then, the development of data-acquisition tools and interpretation techniques has continued unabated1–19. Now, dynamic dysfunction data is collected down-hole using measurement-while drilling (MWD) systems and transmitted to the rig floor in near real time2–6. In fact, recently, some of the tools have become available as stand-alone subs to address those applications in which a complete MWD system could not be justified7–9. Even so, the tools have at least three limitations when characterizing bit behavior is the objective:the cost can be prohibitive which can preclude the use of a tool even when dynamic dysfunction is suspected,the data-acquisition tool may be removed from the bit by as much as 10 - 100 feet and therefore, the data collected may not necessarily represent the conditions at the bit andthe subbased tools alter the bottom-hole assembly.