We develop a unified Petrov-Galerkin spectral method for a class of fractional partial differential equations with two-sided derivatives and constant coefficients of the form 0D t u+ ∑d i=1[cli aiD 2μi xi u+ cri xiD 2μi bi u] +γ u = ∑d j=1[ κlj ajD 2νj xj u+κrj xjD 2νj bj u] +f , where 2τ ∈ (0, 2), 2μj ∈ (0, 1) and 2νj ∈ (1, 2), in a (1 +d)-dimensional space-time hypercube, d = 1, 2, 3, · · · ,...

Systems biology of plants offers myriad opportunities and many challenges in modeling. A number of technical challenges stem from paucity of computational methods for discovery of the most fundamental properties of complex dynamical systems. In systems engineering, eigen-mode analysis have proved to be a powerful approach. Following this philosophy, we introduce a new theory that has the benefi...

An universality theorem on the approximation of analytic functions by shifts ζ(s+iτ, F ) of zeta-functions of normalized Hecke-eigen forms F , where τ takes values from the set {kαh : k = 0, 1, 2, . . . } with fixed 0 < α < 1 and h > 0, is obtained.

We study the asymptotic behavior of a singular potential that arises under several frequently occurring analytic behaviors of the eigen functions (of the Schrödinger eigenvalue problem) without introducing cut-offs. Instead, in our analyses we focus on power behaviors of eigen functions. We find that the asymptotic behavior of the singular potential crucially depends on the analytic property of...

Eigen-functions are of key importance in graph mining since they can be used to approximate many graph parameters, such as node centrality, epidemic threshold, graph robustness, with high accuracy. As real-world graphs are changing over time, those parameters may get sharp changes correspondingly. Taking virus propagation network for example, new connections between infected and susceptible peo...

In a three-dimensional context, it is shown how wave functions may be chosen from among the solutions of the Schrodinger equation in a consistent way without demanding single valuedness, but by imposing an invariance condition on the domain of certain given linear operators. This leads to multiple-valued wave functions in certain situations, but disallows unquantized monopoles and "unusual" ang...

This paper is devoted to the study of eigen-sequences for some important operators acting on sequences. Using functional equations involving generating functions, we completely solve the problem of characterizing the fixed sequences for the Generalized Binomial operator. We give some applications to integer sequences. In particular we show how we can generate fixed sequences for Generalized Bin...

The Lam and Rott theory of linearized unsteady boundary layers is revisited, and some new results are obtained. The exact outer eigensolution for a flat plate found in the original paper is shown to be a special case of the Prandtl transposition theorem. The streamwise coordinate-dependent factor of the inner eigen-solutions, first found by M. E. Goldstein for the flat plate, is generalized for...

We use a path integral representation to solve the Eigen and Crow-Kimura molecular evolution models for the case of multiple fitness peaks with arbitrary fitness and degradation functions. In the general case, we find that the solution to these molecular evolution models can be written as the optimum of a fitness function, with constraints enforced by Lagrange multipliers and with a term accoun...

Eigen-channel approach to heavy-ion fusion reactions is exact only when the excitation energy of the intrinsic motion is zero. In order to take into account effects of finite excitation energy, we introduce an energy dependence to weight factors in the eigen-channel approximation. Using two channel problem, we show that the weight factors are slowly changing functions of incident energy. This s...