On the zeros of exponential polynomials

Suppose r = (r₁, …, r_M), r_j ? 0, γ_kj ? 0 integers, k = 1, 2, …, N, j = 1, 2, …, M, γ_k · r = ∑_jγ_kjr_j. The purpose of this paper is to study the behavior of the zeros of the function h(λ, a, r) = 1 + ∑_{j = 1}^Na_je^{?λγ_j · r}, where each a_j is a nonzero real number. More specifically, if $\text{Z}\text{?}\text{(a, r) =}\text{closure}\text{{}\text{Re}\text{λ: h(λ, a, r) = 0}}$, we study the dependence of $\text{Z}\text{?}\text{(a, r)}\text{on}\text{a, r}$. This set is continuous in a but generally not in r. However, it is continuous in r if the components of r are rationally independent. Specific criterion to determine when $\text{0 ?}\text{Z}\text{?}\text{(a, r)}$ are given. Several examples illustrate the complicated nature of $\text{Z}\text{?}\text{(a, r)}$. The results have immediate implication to the theory of stability for difference equations x(t) ? ∑_{k = 1}^MA_kx(t ? r_k) = 0, where x is an n-vector, since the characteristic equation has the form given by h(λ, a, r). The results give information about the preservation of stability with respect to variations in the delays. The results also are fundamental for a discussion of the dependence of solutions of neutral differential difference equations on the delays. These implications will appear elsewhere.     

A general dynamic synthesis for structures with discrete substructures

This paper presents a substructure synthesis method for the dynamic simulation of complex structures, where the structures consist of an assemblage of discrete substructures. An analogy between distributed and discrete structures is extensively invoked. To stimulate the motion of discrete substructures, the concept of “admissible vectors” is introduced, where admissible vectors represent the discrete counterpart of admissible functions for distributed substructures. The individual substructures are forced to act as a whole structure by imposing certain geometric compatibility on internal boundaries shared by any two substructures. A numerical example illustrating the method is presented.     

Enantioselective formal total synthesis of the Dendrobatidae frog toxin, (+)-pumiliotoxin B, via O-directed alkyne free radical hydrostannation

Herein we describe our application of the O-directed free radical hydrostannation of disubstituted alkylacetylenes (with Ph₃SnH and Et₃B) to the (+)-pumiliotoxin B total synthesis problem. Specifically, we report on the use of this method in the synthesis of the Overman alkyne 8, and thereby demonstrate the great utility of this process in a complex natural product total synthesis setting for the very first time. We also report here on a new, stereocontrolled, and highly practical enantioselective pathway to Overman’s pyrrolidine epoxide partner 9 for 8, which overcomes the previous requirement for use of preparative HPLC to separate the 1:1 mixture of diastereomeric epoxides that was obtained in the original synthesis of 9.     

Treatment in Swelling Solutions Modifying Cellulose Fiber Reactivity – Part 1: Accessibility and Sorption

Of prime importance in reactions involving insoluble cellulosic fibers is the sorption of reagents, which is governed by their degrees of accessibility in substrates. Swelling treatments of cellulosics in alkali solutions alter substrate accessibility leading to changes in their reactivity. In this paper, the first of a two-part series, we collate and examine the results from various studies involving different techniques to characterize modifications in cellulosic fibers after swelling treatments in alkali solutions. Results from measurements of structure and accessibility in fibers with techniques such as water retention, inverse size exclusion chromatography (ISEC), iodine sorption, fiber diameters, and fiber-splitting propensities indicate that the influence of swelling treatments on fiber structure/accessibility is differs with alkali type. The results show that a non-uniform rather than uniform distribution of reagents within structures is a more accurate representation of reactions involving swollen cellulosic fibers. Hence, the observed changes in cellulose-fiber reactivity are governed by the degrees of fiber swelling, and reagent sorption and accessibility during swelling treatments.     

A Modified Poincaré Method for the Persistence of Periodic Orbits and Applications

This paper is devoted to the persistence of periodic orbits under perturbations in dynamical systems generated by evolutionary equations, which are not smoothing in finite time, but only asymptotically smoothing. When the periodic orbit of the unperturbed system is non-degenerate, we show the existence and uniqueness of a periodic orbit (with a minimal period near the minimal period of the unperturbed problem) by using “modified” Poincaré methods. Examples of applications, including the perturbed hyperbolic Navier–Stokes equations, systems of damped wave equations and the system of second grade fluids, are given.     

Slow-motion manifolds,dormant instability,and singular perturbations

If the coexistence of two phases at the transition temperature is kept under observation for a long time, then one observes that the system is not exactly in equilibrium and a very slow evolution driven by surface tension is taking place. Theoretically, one should eventually see a spatially homogeneous state, but the time for settling down is so long that what one actually observes is motion towards a stable state. The complexity of the spatial distribution of the two phases keeps decreasing but appears to be stable for very long periods of time with intermittent periods of fast motion when there are small inclusions of one of the two regions embedded in the other phase. For a simple reaction diffusion model, it is shown that this phenomenon can be explained by investigating the flow on the attractor and the unstable manifolds of equilibria.     

Square and Pulse Waves with Two Delays

This work is concerned with the effects on the dynamics of a differential difference equation with two delays as the delays become unbounded in a fixed direction. This leads to a singularly perturbed delay differential equation with singular parameter and delays (1, d). We study in detail d=2 for the case when =0 yields the Hénon map. In a neighborhood of a generic period doubling point for the Hénon map, we show that there can be either a stable square wave or an unstable pulse wave even though the period two point for the map is always stable.