1,881 research outputs found
Halanay-type theory in the context of evolutionary equations with time-lag
We consider extensions and modifications of a theory due to Halanay, and the context in which such results may be applied. Our emphasis is on a mathematical framework for Halanay-type analysis of problems with time lag and simulations using discrete versions or numerical formulae. We present selected (linear and nonlinear, discrete and continuous) results of Halanay type that can be used in the study of systems of evolutionary equations with various types of delayed argument, and the relevance and application of our results is illustrated, by reference to delay-differential equations, difference equations, and methods
Concerning periodic solutions to non-linear discrete Volterra equations with finite memory
In this paper we discuss the existence of periodic solutions of discrete (and discretized) non-linear Volterra equations with finite memory. The literature contains a number of results on periodic solutions of non linear Volterra integral equations with finite memory, of a type that arises in biomathematics. The “summation” equations studied here can arise as discrete models in their own right but are (as we demonstrate) of a type that arise from the discretization of such integral equations. Our main results are in two parts: (i) results for discrete equations and (ii) consequences for quadrature methods applied to integral equations. The first set of results are obtained using a variety of fixed point theorems. The second set of results address the preservation of properties of integral equations on discretizing them. An expository style is adopted and examples are given to illustrate the discussion
Fixed point theroms and their application - discrete Volterra applications
The existence of solutions of nonlinear discrete Volterra equations is established. We define discrete Volterra operators on normed spaces of infinite sequences of finite-dimensional vectors, and present some of their basic properties (continuity, boundedness, and representation). The treatment relies upon the use of coordinate functions, and the existence results are obtained using fixed point theorems for discrete Volterra operators on infinite-dimensional spaces based on fixed point theorems of Schauder, Rothe, and Altman, and Banach’s contraction mapping theorem, for finite-dimensional spaces
Existence theory for a class of evolutionary equations with time-lag, studied via integral equation formulations
In discussions of certain neutral delay differential equations in Hale’s form, the relationship of the original problem with an integrated form (an integral equation) proves to be helpful in considering existence and uniqueness of a solution and sensitivity to initial data. Although the theory is generally based on the assumption that a solution is continuous, natural solutions of neutral delay differential equations of the type considered may be discontinuous. This difficulty is resolved by relating the discontinuous solution to its restrictions on appropriate (half-open) subintervals where they are continuous and can be regarded as solutions of related integral equations. Existence and unicity theories then follow. Furthermore, it is seen that the discontinuous solutions can be regarded as solutions in the sense of Caratheodory (where this concept is adapted from the theory of ordinary differential equations, recast as integral equations)
Neutral delay differential equations in the modelling of cell growth
In this contribution, we indicate (and illustrate by example) roles that may be played by neutral delay differential equations in modelling of certain cell growth phenomena that display a time lag in reacting to events. We explore, in this connection, questions involving the sensitivity analysis of models and related mathematical theory; we provide some associated numerical results
On some aspects of casual and neutral equations used in mathematical modelling
The problem that motivates the considerations here is the construction of mathematical models of natural phenomena that depend upon past states. The paper divides naturally into two parts: in the first, we expound the inter-connection between ordinary differential equations, delay differential equations, neutral delay-differential equations and integral equations (with emphasis on certain linear cases). As we show, this leads to a natural hierarchy of model complexity when such equations are used in mathematical and computational modelling, and to the possibility of reformulating problems either to facilitate their numerical solution or to provide mathematical insight, or both. Volterra integral equations include as special cases the others we consider. In the second part, we develop some practical and theoretical consequences of results given in the first part. In particular, we consider various approaches to the definition of an adjoint, we establish (notably, in the context of sensitivity analysis for neutral delay-differential equations) roles for well-defined ad-joints and ‘quasi-adjoints’, and we explore relationships between sensitivity analysis, the variation of parameters formulae, the fundamental solution and adjoints
From climate change to pandemics: decision science can help scientists have impact
Scientific knowledge and advances are a cornerstone of modern society. They
improve our understanding of the world we live in and help us navigate global
challenges including emerging infectious diseases, climate change and the
biodiversity crisis. For any scientist, whether they work primarily in
fundamental knowledge generation or in the applied sciences, it is important to
understand how science fits into a decision-making framework. Decision science
is a field that aims to pinpoint evidence-based management strategies. It
provides a framework for scientists to directly impact decisions or to
understand how their work will fit into a decision process. Decision science is
more than undertaking targeted and relevant scientific research or providing
tools to assist policy makers; it is an approach to problem formulation,
bringing together mathematical modelling, stakeholder values and logistical
constraints to support decision making. In this paper we describe decision
science, its use in different contexts, and highlight current gaps in
methodology and application. The COVID-19 pandemic has thrust mathematical
models into the public spotlight, but it is one of innumerable examples in
which modelling informs decision making. Other examples include models of storm
systems (eg. cyclones, hurricanes) and climate change. Although the decision
timescale in these examples differs enormously (from hours to decades), the
underlying decision science approach is common across all problems. Bridging
communication gaps between different groups is one of the greatest challenges
for scientists. However, by better understanding and engaging with the
decision-making processes, scientists will have greater impact and make
stronger contributions to important societal problems
Structure of Dengue Virus: Implications for Flavivirus Organization, Maturation, and Fusion
The first structure of a flavivirus has been determined by using a combination of cryoelectron microscopy and fitting of the known structure of glycoprotein E into the electron density map. The virus core, within a lipid bilayer, has a less-ordered structure than the external, icosahedral scaffold of 90 glycoprotein E dimers. The three E monomers per icosahedral asymmetric unit do not have quasiequivalent symmetric environments. Difference maps indicate the location of the small membrane protein M relative to the overlaying scaffold of E dimers. The structure suggests that flaviviruses, and by analogy also alphaviruses, employ a fusion mechanism in which the distal barrels of domain II of the glycoprotein E are inserted into the cellular membrane
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