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Traces of Matrix Products
No full textGiven two noncommuting matrices, A and B, it is well known that AB and BA have the same trace. This extends to cyclic permutations of products of Aâs and Bâs. It is shown here that for 2Ã2 matrices A and B, whose elements are independent random variables with standard normal distributions, the probability that Tr(ABAB) > Tr(AB) is exactly 1/\sqrt{2}
On the rank-k numerical range of matrices polynomials
No full textThis article introduces the notion of the rank- numerical range of a matrix polynomial , whose coefficients are complex matrices. Also, geometric properties are obtained, including the relation to the ordinary numerical range
A Non-Invasive Population Study of Moose in Northern Yellowstone National Park
No full textNorth American moose (Alces alces) populations are declining across much of their southern distribution from the Canadian Maritimes to the Rocky Mountain range of the Shiras moose subspecies (Alces alces shirasi). Shiras moose population declines have been documented in Montana and Wyoming with reduced productivity reported from Utah and Colorado. These declines are due to a combination of factors including the natural succession, loss, and degradation of habitat; predation by wolves and bears; disease caused by infection from artery worm; and parasitism by moose ticks. The effects of heat stress may also contribute to chronic malnutrition and a reduction in female fertility. Significant reductions in Montana and Wyoming moose populations adjacent to Yellowstone National Park (YNP) are indicative of regional moose population declines and suggest that moose numbers may be decreasing in YNP as well. In the northern portion of YNP, also known as the Northern Range (NR), significant loss of riparian willow browse due to overgrazing by elk for decades and by bison more recently, the reduction of mature and old-growth conifer forests from the fires of 1988 and from disease more recently have reduced both winter habitat quality and quantity for moose. Several moose with cropped ears, an external sign of the disease Elaeophorosis, or artery worm, have been observed over the past few years on the NR suggesting that the disease may also be present in YNP. Northern Range estimates of moose have decreased from almost 400 in 1970 to possibly fewer than 100 today. Despite evidence suggesting moose decline in YNP, no current population data exists. Knowledge of population demographics serves as a critical baseline for evaluating and understanding factors leading to population declines. However, because moose are solitary, prefer densely vegetated habitats, and are present at low densities, collecting population data is challenging. Traditional methods of studying moose that require capture, radio-collaring, and aerial surveys are costly, sometimes produce unreliable results, can be harmful to the study animal, and are discouraged in some jurisdictions such as national parks. Non-invasive sampling, the collection of data without having to capture, handle, or in any manner physically restrain study animals, has proven to be a valuable tool for acquiring accurate population data from free-ranging ungulates when using traditional methods is neither feasible nor practical. In December 2013, we initiated a three-year non-invasive moose population study in YNP with the main objective to estimate population demographics of NR moose. For three consecutive winters we will be systematically collecting fecal pellets from the extent of NR moose wintering habitat. We are extracting DNA from epithelial cells on the pellet surface and through genetic testing will be able to identify individual moose and their genders. Female pellet samples will be analyzed for pregnancy hormone concentrations to make inferences on pregnancy rates. Because fecal pellet size is directly related to moose size, and therefore to moose age, we will use various pellet measurements to differentiate between age classes. These data will be used in capture-recapture modeling to estimate population abundance and vital rates
Field Research and Conservation Class: Clayton High School, Clayton MO
No full textField Research & Conservation emphasizes long-term field research experiences, examines ecosystem processes, and investigates the evolution of American perspectives about nature. Our time spent at the UW-NPS research station is divided between pursuing behavioral ecology research and exploring Grand Teton National Park and the surrounding area. These experiences help students gain understanding of how the region was shaped by geological, biological, and political processes that have been influenced by Americaâs evolving conservation ethic. After the summer field experience, my students assist in data analysis and the development of a poster project. We present our findings at the Phi Sigma Research Symposium at Illinois State University and the St. Louis Ecology, Evolution, and Conservation Retreat at Southern Illinois University Edwardsville
