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Reverse Jensen-Mercer Type Operator Inequalities
No full textLet be a selfadjoint operator on a Hilbert space with spectrum in an interval and be a unital positive linear map, where is also a Hilbert space. Let with $
Signed graphs with small positive index of inertia
No full textIn this paper, the signed graphs with one positive eigenvalue are characterized, and the signed graphs with pendant vertices having exactly two positive eigenvalues are determined. As a consequence, the signed trees, the signed unicyclic graphs and the signed bicyclic graphs having one or two positive eigenvalues are characterized
On the reduction of matrix polynomials to Hessenberg form
No full textIt is well known that every real or complex square matrix is unitarily similar to an upper Hessenberg matrix. The purpose of this paper is to provide a constructive proof of the result that every square matrix polynomial can be reduced to an upper Hessenberg matrix, whose entries are rational functions and in special cases polynomials. It will be shown that the determinant is preserved under this transformation, and both the finite and infinite eigenvalues of the original matrix polynomial can be obtained from the upper Hessenberg matrix
Graphs with reciprocal eigenvalue properties
No full textIn this paper, only simple graphs are considered. A graph G is nonsingular if its adjacency matrix A(G) is nonsingular. A nonsingular graph G satisfies reciprocal eigenvalue property (property R) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and G satisfies strong reciprocal eigenvalue property (property SR) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and they both have the same multiplicities. From the definitions property SR implies property R. Furthermore, for some classes of graphs (for example, trees), it is known that these properties are equivalent. However, the equivalence of these two properties is not yet known for any nonsingular graph. In this article, it is shown that these properties are not equivalent in general
Polynomial reconstruction of signed graphs whose least eigenvalue is close to -2
No full textThe polynomial reconstruction problem for simple graphs has been considered in the literature for more than forty years and is not yet resolved except for some special classes of graphs. Recently, the same problem has been put forward for signed graphs. Here, the reconstruction of the characteristic polynomial of signed graphs whose vertex-deleted subgraphs have least eigenvalue greater than is considered
Automorphisms of a commuting graph of rank one upper triangular matrices
No full textLet be a finite field, an arbitrary integer, the set of all matrices over , and the set of all rank one upper triangular matrices of order . For , denote . The commuting graph of , denoted by , is the simple undirected graph with vertex set in which for every two distinct vertices and , is an edge if and only if . In this paper, it is shown that any graph automorphism of with can be decomposed into the product of an extremal automorphism, an inner automorphism, a field automorphism and a local scalar multiplication
Lorentz transformation from an elementary point of view
No full textElementary methods are used to examine some nontrivial mathematical issues underpinning the Lorentz transformation. Its eigen-system is characterized through the exponential of a -skew symmetric matrix, underlining its unconnectedness at one of its extremes (the hyper-singular case). A different yet equivalent angle is presented through Pauli coding which reveals the connection between the hyper-singular case and the shear map
Boom and bust of an aquatic invasive species? Population monitoring of the New Zealand mudsnail (Potamopyrgus antipodarum) and interactions with native species in Polecat Creek, WY
No full textThe invasive New Zealand mudsnail (Potamopyrgus antipodarum) has been found to reach densities exceeding 500,000 individuals/m2 in Polecat Creek, located in the Greater Yellowstone Ecosystem in Wyoming. The biomass of P. antipodarum in Polecat Creek has declined in recent years, suggesting the population “boomed and busted”; the population was booming in 2000-2001, but in 2011 the biomass had decreased by ~93%, suggesting a “bust” period for P. antipodarum. Native, net-spinning caddisflies (Hydropsyche spp.) have increased dramatically in biomass from 2001-2010, which may indicate that some native macroinvertebrates have increased in biomass due to release of suppression by P. antipodarum. I collected macroinvertebrate core samples in Polecat Creek to monitor any changes in macroinvertebrate biomass and performed field experiments to determine a possible mechanism by which P. antipodarum may have suppressed Hydropsyche caddisfly populations. I allowed Hydropsyche larvae to establish and build nets on tiles within experimental chambers in Polecat Creek and added “boom” and “bust” densities of P. antipodarum to chambers. Preliminary results showed no significant difference between the number of nets present in control chambers excluding P. antipodarum and chambers containing “boom” and “bust” densities of P. antipodarum. This suggests that P. antipodarum do not actively destroy nets, but may interfere with feeding by clustering upon nets.
Featured photo from Figure 2 in report
