Mathematics

We prove that all the zeros of a certain family of meromorphic functions are on the critical line Re(s) = 1/2, and are simple (except possibly when s = 1/2). We prove this by relating the zeros to the discrete spectrum of unbounded self-adjoint operators. For example, for h(s) a meromorphic function with no zeros in Re(s) > 1/2, real-valued on R, and h(1−s) h(s) |s| 1− in Re(s) > 1/2, the only zeros of h(s) ± h(1 − s) are on the critical line. More generally, we can allow certain patterns of finitely-many zeros and poles in the right half-plane. One instance of such a function h is h(s) = ξ k (2s), the completed zeta-function of a number field k, or, more generally, many self-dual automorphic L-functions. We use spectral theory suggested by results of Lax-Phillips and ColinDeVerdière. This simplifies ideas of W.

Physicists such as Green, Vanhove, et al show that differential equations involving automorphic forms govern the behavior of gravitons. One particular point of interest is solutions to (∆−λ)u = E α E β on an arithmetic quotient of the exceptional group E 8 . We establish that the existence of a solution to (∆ − λ)u = E α E β on the simpler space SL 2 (Z)\SL 2 (R) for certain values of α and β depends on nontrivial zeros of the Riemann zeta function ζ(s). Further, when such a solution exists, we use spectral theory to solve (∆ − λ)u = E α E β on SL 2 (Z)\SL 2 (R) and provide proof of the meromorphic continuation of the solution. The construction of such a solution uses Arthur truncation, the Maass-Selberg formula, and automorphic Sobolev spaces. 7

Shannon entropy is used to provide an estimate of the number of interpretable components in a principal component analysis. In addition, several ad hoc stopping rules for dimension determination are reviewed and a modification of the broken stick model is presented. The modification incorporates a test for the presence of an "effective degeneracy" among the subspaces spanned by the eigenvectors of the correlation matrix of the data set then allocates the total variance among subspaces. A summary of the performance of the methods applied to both published microarray data sets and to simulated data is given.

A particular interaction-diffusion mussel-algae model system for the development of spontaneous stationary young mussel bed patterning on a homogeneous substrate covered by a quiescent marine layer containing algae as a food source is investigated employing weakly nonlinear diffusive instability analyses. The main results of these analyses can be represented by plots in the ratio of mussel motility to algae lateral diffusion versus the algae reservoir concentration dimensionless parameter space. Regions corresponding to bare sediment and mussel patterns consisting of rhombic or hexagonal arrays and isolated clusters of clumps or gaps, an intermediate labyrinthine state, and homogeneous distributions of low to high density may be identified in this parameter space. Then those Turing diffusive instability predictions are compared with both relevant field and laboratory experimental evidence and existing numerical simulations involving differential flow migrating band instabilities for the associated interaction-dispersion-advection mussel-algae model system as well as placed in the context of the results from some recent nonlinear pattern formation studies.

Equine Infectious Anemia Virus (EIAV) is a retrovirus that establishes a persistent infection in horses and ponies. The virus is in the same lentivirus subgroup that includes human immunodeficiency virus (HIV). The similarities between these two viruses make the study of the immune response to EIAV relevant to research on HIV. We developed a mathematical model of within-host EIAV infection dynamics that contains both humoral and cell-mediated immune responses. Analysis of the model yields results on thresholds that would be necessary for a combined immune response to successfully control infection. Numerical simulations are presented to illustrate the results. These findings have the potential to lead to immunological control measures for lentiviral infection.

A rhombic planform nonlinear cross-diffusive instability analysis is applied to a particular interaction-diffusion plant-ground water model system in an arid flat environment. This model contains a plant root suction effect as a cross-diffusion term in the ground water equation. In addition a threshold-dependent paradigm that differs from the usually employed implicit zero-threshold methodology is introduced to interpret stable rhombic patterns. These patterns are driven by root suction since the plant equation does not yield the required positive feedback necessary for the generation of standard Turing-type self-diffusive instabilities. The results of that analysis can be represented by plots in a root suction coefficient versus rainfall rate dimensionless parameter space. From those plots regions corresponding to bare ground and vegetative patterns consisting of isolated patches, rhombic arrays of pseudo spots or gaps separated by an intermediate rectangular state, and homogeneous distributions from low to high density may be identified in this parameter space. Then, a morphological sequence of stable vegetative states is produced upon traversing an experimentally-determined root suction characteristic curve as a function of rainfall through these regions. Finally, that predicted sequence along a rainfall gradient is compared with observational evidence relevant to the occurrence of leopard bush, pearled bush, or labyrinthine tiger bush vegetative patterns, used to motivate an aridity classification scheme, and placed in the context of some recent biological nonlinear pattern formation studies.

An existing weakly nonlinear diffusive instability hexagonal planform analysis for an interactiondiffusion plant-surface water model system in an arid flat environment [11] is extended by performing a rhombic planform analysis as well. In addition a threshold-dependent paradigm that differs from the usually employed implicit zero-threshold methodology is introduced to interpret stable rhombic patterns. The results of that analysis are synthesized with those of the existing hexagonal planform analysis. In particular these synthesized results can be represented by closedform plots in the rate of precipitation versus the specific rate of plant density loss parameter space. From those plots, regions corresponding to bare ground and vegetative Turing patterns consisting of tiger bush (parallel stripes and labyrinthine mazes), pearled bush (hexagonal gaps and rhombic pseudo-gaps), and homogeneous distributions of vegetation, respectively, may be identified in this parameter space. Then that predicted sequence of stable states along a rainfall gradient is both compared with observational evidence and used to motivate an aridity classification scheme. Finally this system is shown to be isomorphic to the chemical reaction-diffusion Gray-Scott model and that isomorphism is employed to draw some conclusions about sideband instabilities as applied to vegetative patterning.

Analysis of previously published target-cell limited viral dynamic models for pathogens such as HIV, hepatitis, and influenza generally rely on standard techniques from dynamical systems theory or numerical simulation. We use a quasi-steady-state approximation to derive an analytic solution for the model with a non-cytopathic effect, that is, when the death rates of uninfected and infected cells are equal. The analytic solution provides time evolution values of all three compartments of uninfected cells, infected cells, and virus. Results are compared with numerical simulation using clinical data for equine infectious anemia virus, a retrovirus closely related to HIV, and the utility of the analytic solution is discussed.

Shannon entropy is used to provide an estimate of the number of interpretable components in a principal component analysis. In addition, several ad hoc stopping rules for dimension determination are reviewed and a modification of the broken stick model is presented. The modification incorporates a test for the presence of an "effective degeneracy" among the subspaces spanned by the eigenvectors of the correlation matrix of the data set then allocates the total variance among subspaces. A summary of the performance of the methods applied to both published microarray data sets and to simulated data is given.

The optimization of large trusses often leads to a nearly optimal solution, rather than a truly optimal design. In fact, the problem space for truss optimization grows exponentially with the size of the truss. Using the method of problem reduction, this paper demonstrates that truss optimization is in the set of NP-complete problems. Hence, the only practical techniques for solving the truss problem are heuristic in nature. Genetic algorithms provide a viable solution for large trusses.

Let p be a prime number with p≠2. We consider second order linear recurrence relations of the form S n =aS n-1 +bS n-2 over the finite field Z p (we assume b≠0). Results regarding the period and distribution of elements in the sequence {S 0 ,S 1 ,...} are well-known (see works by Kuipers, Niederreiter, Wall, and Webb). We examine these second order recurrences using matrices, groups, and G-sets.

Second order linear homogeneous recurrence relations with coefficients in a finite field or in the integers modulo of an ideal have been the subject of much study (see for example [1, 2, 4, 5, 6, 7, 8, 9]). This paper extends many of these results to finite rings. In the first part of this paper we develop polynomials which generate purely periodic sequences over any finite ring, R. We then use these polynomials with coefficients in R to establish bounds on the period of these sequences.

In an article published in 1979, Kainen and Bernhart [1] laid the groundwork for further study of book embeddings of graphs. They define an $n$-book as a line $L$ in 3-space, called the spine, and $n$ half-planes, called pages, with $L$ as their common boundary. An $n$-book embedding of a graph $G$ is an embedding of $G$ in an $n$-book so that the vertices of $G$ lie on the spine and each edge of $G$ lies within a single page so that no two edges cross. The book thickness $bt(G)$ or page number $pg(G)$ of a graph $G$ is the smallest $n$ so that $G$ has an $n$-book embedding. Finding the book thickness of an arbitrary graph is a difficult problem. Even with a pre-specified vertex ordering, the problem has been shown to be NP-complete [6]. In this paper we will introduce book embeddings with particular focus on results for graphs with small book thickness.

Let p be a prime number with p = 2. We consider sequences generated by nth order linear recurrence relations over the finite field Zp. In the first part of this paper we generalize some of the ideas in [6] to nth order linear recurrences. We then consider the case where the characteristic polynomial of the recurrence has one root in Zp of multiplicity n. In this case, we show that the corresponding recurrence can be generated by a relatively simple matrix.

A graph is called dispersable if it has a book embedding in which each page has maximum degree 1 and the number of pages is the maximum degree. Bernhart and Kainen conjectured every k-regular bipartite graph is dispersable. Forty years later, Alam, Bekos, Gronemann, Kaufmann, and Pupyrev have disproved this conjecture, identifying nonplanar 3and 4-regular bipartite graphs that are not dispersable. They also proved all cubic planar bipartite 3-connected graphs are dispersable and conjectured that the connectivity condition could be relaxed. We prove that every cubic planar bipartite multigraph is dispersable. Key Phrases: book thickness; dispersable book embeddings; matching book thickness; subhamiltonian vertex order; cubic bipartite planar multigraphs.