The eight Cayley-Dickson doubling product

The purpose of this paper is to identify all eight of the basic Cayley-Dickson doubling products. A Cayley-Dickson algebra \cda{N+1} of dimension $2^{N+1}$ consists of all ordered pairs of elements of a Cayley-Dickson algebra \cda{N} of dimension $2^N$ where the product $(a,b)(c,d)$ of elements of \cda{N+1} is defined in terms of a pair of second degree binomials $\left(f(a,b,c,d),g(a,b,c,d)\right)$ satisfying certain properties. The polynomial pair$(f,g)$ is called a `doubling product.' While \cda{0} may denote any ring, here it is taken to be the set $\mathbb{R}$ of real numbers. The binomials $f$ and $g$ should be devised such that \cda{1}=\mathbb{C} the complex numbers, \cda{2}=\mathbb{H} the quaternions, and \cda{3}=\mathbb{O} the octonions. Historically, various researchers have used different yet equivalent doubling products.Comment: 32 candidates for alternate Cayley-Dickson doubling products are winnowed down to 8 products. Author now finds that 4 of those 8 should also be discarded: each allows zero divisors at the eight dimensional stage. The 4 remaining products are denoted in the paper as P0,P3(the standard doubling product),P4,P7. Those four produce algebras isomorphic to the standard Cayley-Dickson algebra

Marking parts to aid robot vision

The premarking of parts for subsequent identification by a robot vision system appears to be beneficial as an aid in the automation of certain tasks such as construction in space. A simple, color coded marking system is presented which allows a computer vision system to locate an object, calculate its orientation, and determine its identity. Such a system has the potential to operate accurately, and because the computer shape analysis problem has been simplified, it has the ability to operate in real time

Perceived Indicators of Support Leading to the Successful Attainment of the National Board for Professional Teaching Standards Certification.

There has been limited research completed to identify supports or supporting behaviors that assisted candidates while they were completing the national board certification process. Identifying support factors that assisted national board certified teachers to the successful completion of the process was the primary focus of this study. A qualitative research method was used to interview national board certified teachers, their teaching colleagues, and their administrators. Questions were posed from the interview guides developed for each of the three targeted groups. Information derived from the interviews focused on the perceived supports that lead to the successful attainment of the national board certification. Each interview was audio taped and transcribed verbatim and then systematically coded and analyzed. The thick and rich descriptions provided an opportunity for professionals to identify with the study\u27s participants and therefore established the applicability of the study. The specific findings were organized and clustered around the research questions. Subtopics that emerged from the data analysis process were also addressed as they related to the five researched topics. The information that accumulated prompted the formation of specific recommendations. This study should be of interest to the National Board for Professional Teaching Standards (NBPTS) as they continue to revisit their set of rigorous standards and the certification process. It should also be of interest to prospective teachers seeking the certification along with their administrators as supporters. The study could also be of interest to school system administrators who seek to promote staff development experiences for their teachers or to those individuals who desire to assist teachers in becoming highly qualified. Prompted by the No Child Left Behind legislation, our nation began to focus on the effects of teacher quality and students\u27 achievement. The national board certification provided by the NBPTS is one means of meeting the criteria set forth to be deemed highly qualified

Evapotranspiration Mapping for Forest Management in California's Sierra Nevada

We assessed the response of densely forested watersheds with little apparent annual water limitation to forest disturbance and climate variability, by studying how past wildfires changed forest evapotranspiration, and what past evapotranspiration patterns imply for the availability of subsurface water storage for drought resistance. We determined annual spatial patterns of evapotranspiration using a top-down statistical model, correlating measured annual evapotranspiration from eddycovariance towers across California with NDVI (Normalized Difference Vegetation Index) measured by satellite, and with annual precipitation. The study area was the Yuba and American River watersheds, two densely forested watersheds in the northern Sierra Nevada. Wildfires in the 1985-2015 period resulted in significant post-fire reductions in evapotranspiration for at least 5 years, and in some cases for more than 20 years. The levels of biomass removed in medium-intensity fires (25- 75% basal area loss), similar to magnitudes expected from forest treatments for fuels reduction and forest health, reduced evapotranspiration by as much 150-200 mm yr-1 for the first 5 years. Rates of recovery in post-wildfire evapotranspiration confirm the need for follow-up forest treatments at intervals of 5-20 years to sustain lower evapotranspiration, depending on local landscape attributes and interannual climate. Using the metric of cumulative precipitation minus evapotranspiration (P-ET) during multi-year dry periods, we found that forests in the study area showed little evidence of moisture stress during the 1985-2018 period of our analysis, owing to relatively small reliance on interannual subsurface water storage to meet dry-year evapotranspiration needs of vegetation. However, more-severe or sustained drought periods will push some lower-elevation forests in the area studied toward the cumulative P-ET thresholds previously associated with widespread forest mortality in the southern Sierra Nevada

The weld-brazing metal joining process

Superior mechanical properties were obtained in metal joints weld-brazed between faying surfaces. Weld-braze applications and advantages are listed

Weld-brazing - a new joining process

A joining process designated weld brazing which combines resistance spot welding and brazing has been developed. Resistance spot welding is used to position and align the parts as well as to establish a suitable faying surface gap for brazing. Fabrication is then completed by capillary flow of the braze alloy into the joint. The process has been used successfully to fabricate Ti-6Al-4V titanium alloy joints using 3003 aluminum braze alloy. Test results obtained on single overlap and hat-stiffened structural specimens show that weld brazed joints are superior in tensile shear, stress rupture, fatigue, and buckling than joint fabricated by spotwelding or brazing. Another attractive feature of the process is that the brazed joints is hermetically sealed by the braze material