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Effects of wolf and grizzly bear recovery on cougars in the Southern Greater Yellowstone Ecosystem
No full textWithin the large carnivore guild, competitively dominant species can limit the population sizes and alter the behavior of subordinate competitors. However, the mechanisms by which dominant competitors affect subordinates are complex and challenging to disentangle, particularly for free ranging large mammals. For my dissertation, I took advantage of a natural experiment and sixteen years of location data from a subordinate carnivore (cougars), two dominant competitors (wolves and grizzly bears), and a shared prey species (elk), as well as kill site data from cougars in the Southern Greater Yellowstone Ecosystem (SGYE), a multi-use landscape with numerous anthropogenic impacts. My dissertation was an investigation of three mechanisms, both direct and indirect, by which competition from recovering wolves and grizzly bears affected subordinate cougars in a system where human impacts also play strong roles in shaping dynamics. Specifically, I evaluated 1) whether cougar habitat selection changed as wolf and grizzly bear populations recovered and whether these changes could be attributed to cougars actively avoiding dominant competitors; 2) how cougar access to the habitat of its primary prey species was affected by wolf and grizzly bear recovery; and 3) how kleptoparasitism from recovering wolves and grizzly bears – as well as black bears –may have affected the SGYE cougar population.
Featured photo by Yellowstone National Park on Flickr (https://flic.kr/p/2n3TNx8)
Orthogonal Procrustes and norm-dependent optimality
No full textThis note revisits the classical orthogonal Procrustes problem and investigates the norm-dependent geometric behavior underlying Procrustes alignment for subspaces. It presents generic, deterministic bounds quantifying the performance of a specified Procrustes-based choice of subspace alignment. Numerical examples illustrate the theoretical observations and offer additional, empirical findings which are discussed in detail. This note complements recent advances in statistics involving Procrustean matrix perturbation decompositions and eigenvector estimation
On Jacob's construction of the rational canonical form of a matrix
No full textH.G. Jacob's elegant approach to the rational canonical, or Frobenius normal form of a linear map is presented here in pure matrix language, thereby avoiding the abstract machinery and prerequisites in the original paper. Related algorithmic aspects and an efficient implementation in the computer algebra system GAP are also discussed
On the mean and dispersion of the Moore-Penrose generalized inverse of a Wishart matrix
No full textThe Moore-Penrose inverse of a singular Wishart matrix is studied. When the scale matrix equals the identity matrix the mean and dispersion matrices of the Moore-Penrose inverse are known. When the scale matrix has an arbitrary structure no exact results are available. The article complements the existing literature by deriving upper and lower bounds for the expectation and an upper bound for the dispersion of the Moore-Penrose inverse. The results show that the bounds become large when the number of rows (columns) of the Wishart matrix are close to the degrees of freedom of the distribution
On the Critical Ideals of Complete Multipartite Graphs
No full textThe notions of critical ideals and characteristic ideals of graphs are introduced by Corrales and Valencia to study properties of graphs, including clique number, zero forcing number, minimum rank and critical group. In this paper, we provide methods to compute critical ideals of complete multipartite graphs and obtain complete answers for the characteristic ideals of complete multipartite graphs
Rational Criteria for Diagonalizability of Real Matrices
No full textThe purpose of this note is to obtain rational criteria for diagonalizability of real matrices through the analysis of the moment and Gram matrices associated to a given real matrix. These concepts were introduced by Horn and Lopatin in [R.A. Horn and A.K. Lopatin. The moment and Gram matrices, distinct eigenvalues and zeroes, and rational criteria for diagonalizability. Linear Algebra and its Applications, 299:153-163, 1999] for complex matrices. However, when the matrix is real, it is possible to combine their results with the Borchardt-Jacobi Theorem, in order to get new and noteworthy rational criteria
Minors of a skew symmetric matrix: a combinatorial approach
No full textKnuth's combinatorial approach to Pfaffians is used to reprove and clarify a century-old formula, due to Brill. It expresses arbitrary minors of a skew symmetric matrix in terms of Pfaffians.
 
The outer generalized inverse of an even-order tensor
No full textNecessary and sufficient conditions for the existence of the outer inverse of a tensor with the Einstein product are studied. This generalized inverse of a tensor unifies several generalized inverses of tensors introduced recently in the literature, including the weighted Moore-Penrose, the Moore-Penrose, and the Drazin inverses. The outer inverse of a tensor is expressed through the matrix unfolding of a tensor and the tensor folding. This expression is used to find a characterization of the outer inverse through group inverses, establish the behavior of outer inverse under a small perturbation, and show the existence of a full rank factorization of a tensor and obtain the expression of the outer inverse using full rank factorization. The tensor reverse rule of the weighted Moore-Penrose and Moore-Penrose inverses is examined and equivalent conditions are also developed
Determinants of Normalized Bohemian Upper Hessenberg Matrices
No full textA matrix is Bohemian if its elements are taken from a finite set of integers. An upper Hessenberg matrix is normalized if all its subdiagonal elements are ones, and hollow if it has only zeros along the main diagonal. All possible determinants of families of normalized and hollow normalized Bohemian upper Hessenberg matrices are enumerated. It is shown that in the case of hollow matrices the maximal determinants are related to a generalization of Fibonacci numbers. Several conjectures recently stated by Corless and Thornton follow from these results
Equivalent Characterizations of the Spectra of Graphs and Applications to Measures of Distance-regularity
No full textThe spectrum of a graph usually provides a lot of information about its combinatorial structure. Moreover, from the spectrum, the so-called predistance polynomials can be defined, as a generalization, for any graph, of the distance polynomials of a distance-regular graph. Going further, the preintersection numbers generalize the intersection numbers of a distance-regular graph. This paper describes, for any graph, the closed relationships between its spectrum, predistance polynomials, and preintersection numbers. Then, some applications to derive combinatorial properties of the given graph, most of them related to some fundamental characterizations of distance-regularity, are presented. In particular, the so-called `spectral excess theorem' is revisited. This result states that a connected regular graph is distance-regular if and only if its spectral excess, which is a value computed from the spectrum, equals the average excess, that is, the mean of the numbers of vertices at maximum distance from every vertex